We perform a proper-motion pair searching over our sample  of the 3745  M dwarfs/subdwarfs using their Gaia coordinates, proper motions, and parallaxes [Our routine first separates any two stars with a small difference in their  equatorial coordinates, i.e., right ascension and declination. These pairs are then checked based on their parallaxes and proper motion components along right ascension and declination to find common proper-motion pairs. Our inspection confirms that  these pairs are real binary systems]. We identify 130 common proper-motion pairs, most of which   have metallicity estimates in the near-solar metallicity range ($-$0.5$\leq$[M/H]$\leq$+0.5 dex), but with a few systems having metal-poor metal-poor components. Figures 38 and 39 compare the [M/H], [$\alpha$/Fe] and [M/H]+[$\alpha$/Fe] of the primaries with those  of their respective companions, using the normal and “reduced-correlation method”, respectively. For each method, we also examined the parameter values using the two gravity-modeling approaches, i.e., when surface gravity is regarded as a fixed parameter (top panels) and when surface gravity is allowed to vary (bottom panels). In all cases, we exclude any pair with at least one component having T$_\textrm{\footnotesize{eff}}$$>$3550K, whose best-fit estimates may suffer from larger systematic uncertainties.  The standard deviation of  the differences in parameter values associated with the primaries and their companions is shown in each panel.

As can be seen from these figures,   the model-fit values inferred from the constant-gravity approach  generally show a  better agreement for the two components in each binary system, as compared to those obtained from the variable-gravity approach. In addition, the “reduced-correlation method” does not necessarily improve the precision of the resulting best-fit estimates;  there is only a slight difference between the standard deviation of the differences in parameter values from the normal and “reduced-correlation method” for a given gravity-modeling approach. This again indicates that the residual correlations between wavelength datapoints have no considerable effect on the derived parameters  and  systematic issues in modeling stellar atmospheres can  cause  significant  uncertainties in the synthetic fitting using low-resolution spectra. Nevertheless, our measured  parameters are  still sufficiently precise  to present  a reasonable level of chemical homogeneity between the components of the binary systems. Considering its high efficiency,  our pipeline  can therefore be used for large samples of common proper-motion pairs to examine the chemical parameters of their components.

The primary and companion of the  74 pairs along with their inferred classification and chemical parameters [M/H] and [$\alpha$/Fe] from the “normal method” using the constant-gravity approach are listed in Table 3.

§ KINEMATIC ANALYSIS

We calculate  the Galactic velocities U (towards the Galactic center), V (in the direction of Galactic rotation), and  W (towards the north of the Galactic plane) of our stars and then plot the  two-dimension projections on  the UV, VW and UW planes. Although we do derive a radial velocity correction for our stars to find their respective  best-fit models, these velocities are unreliable because the long-slit spectra were collected primarily for spectral classification purposes, and no radial velocity calibration standards were collected, which makes the absolute wavelength calibration of the stars relatively uncertain. In  addition, the radial velocities that are inferred from synthetic fitting are dependent on the atomic and molecular line list used in model atmospheres. We find  some offsets  between the radial velocities inferred from the model fitting using the PHOENIX and MARCS (Gustafsson et al. 2008) model atmospheres that use different sources of line lists. Although such velocity corrections are precise enough to determine best-fit model spectra, they are not reliable to calculate the Galactic velocity components U, V, and W. Moreover, none of  the M dwarf/subdwarfs in our subset have radial velocity from the Gaia catalog (they are too faint for this).

For this reason, we compute these components of motion  to a first approximation assuming radial velocity $v_{r}$=0. The computed  U, V, and W velocities  are shifted to a Galactocentric reference frame assuming that the Sun has a component of motion [11.1 km/s, 12.24 km/s, 7.25 km/s] relative to the local standard of rest (Schönrich et al. 2010), and that the local standard of rest has a component of motion [U, V, W]=[0, +220km/s, 0] relative to the Galactic center. To construct the UV, VW and UW kinematic plots, we use only stars in specific sectors of the sky to minimize the effect of the missing radial velocities[This can be perceived from the formulation of U, V, and W:  U= $v_{r}$cos(l)cos(b)$-$$v_{l}$sin(l)$-$$v_{b}$cos(l)sin(b), V= $v_{r}$sin(l)cos(b)+$v_{l}$cos(l)$-$$v_{b}$sin(l)sin(b), W= $v_{r}$sin(b)+$v_{b}$cos(b), where $v_{r}$ is the radial velocity, l is the Galactic longitude, b is the Galactic latitude, and $v_{l}$ and $v_{b}$ are the tangential velocity components along l and b, respectively. The position of stars on the three sectors of the sky,  with respect to l and b, makes the contribution of the first term that includes $v_{r}$ significantly small, for each of  the UV, VW, and UW cases.]. To this end,  we divide the 3745 stars into three groups  according to  their positions  on three different  sectors of the sky, as described below:

* 1311 stars with positions on the sky closest to the Galactic poles are only used for the UV plots (hereafter,  Group UV).

* 1048 stars with positions closest to the Galactic center and anti-center are only used for the VW plots (hereafter, Group VW).

* 1386 stars with positions closest to the solar apex and anti-apex are used only for the UW plots (hereafter, Group UW).

Figure 40 presents the projected motions in the two-dimensional UV, VW, and UW  diagrams for subsets of the respective Group UV (left panels), VW (middle panels), and UW (right panels), including all stars with  T$_\textrm{\footnotesize{eff}}$$\leq$3550 K and   $-$2.5<[M/H]<+0.5 dex,  color-mapped based on their [M/H] values  inferred from the  “normal method” when surface gravity is a constant parameter.  In all panels, the  metal-poor  stars are well separated both chemically and kinematically  from the   metal-rich  stars,
which is consistent with the metal-poor M subdwarfs ([M/H]<$-$0.5) generally being members of the local Galactic halo population, while the metal-rich M dwarfs ([M/H]$\geq$$-$0.5) are members of the local Galactic disk population.  As expected, the metal-poor stars, that belong to older (halo) stellar populations, show larger dispersions in the Galactic  velocities U, V, and W. On the other hand, the metal-rich stars, that are younger, have smaller dispersions in their velocity components. There is no noticeable  difference in the plots using the model-fit values from the “reduced-correlation method”,  as compared to those in Figure 40, and we do not include them in this paper.

An alternative way to show the contrast between the halo and disk stars in chemodynamical space is to color-map stars in stars in the [$\alpha$/Fe]  versus [M/H] diagram according to their U, V, and W velocity components. For this purpose, we combine  Groups UV and UW (hereafter, Group UV+UW), Groups UV and VW (hereafter, Group UV+VW), and Groups UW and VW (hereafter, Group UW+VW), and then examine the distribution of the combined groups for stars with  T$_\textrm{\footnotesize{eff}}$$\leq$3550 K and   $-$2.5<[M/H]<+0.5 dex in the [$\alpha$/Fe]-[M/H] plane, as shown in Figure 41. The parameter model-fit estimates are those derived from the pipeline where surface gravity is kept fixed and randomized within the corresponding step size in the model grid. The distributions are color-mapped according to U, V, and W velocity components for  Groups UV+UW, UV+VW, and UW+VW, respectively[In order to make a better color map with a clearer contrast between the two populations, i.e., metal-poor and metal-rich stars, for the W velocity components, we exclude 5 stars with W<$-$200 km/s and W>350 km/s.].

The top panel shows  a systematic difference in U velocity  between the  metal-rich and   the metal-poor stars;  the higher-metallicity stars tend to have lower velocity components towards the Galactic center, which is  consistent with the motion of the Galactic disk  stars. On the other hand,   the lower-metallicity stars possess more randomly directed velocities with higher values of U component, which is compatible with the motion of the Galactic halo stars. The middle panel also confirms that the metal-rich stars have velocities that are characteristic  of  the disk  with  higher rotational  velocities, whereas the  metal-poor stars  show  random motions with lower V components. As seen in the bottom panel, the clear contrast in the W components  between the metal-rich and the metal-poor stars is again consistent with the difference between the   motion of  the  disk and halo populations. In  other words,  the metal-rich stars typically have lower velocity components perpendicular to the Galactic plane, which is an indication of the stellar motions associated with  the Galactic disk, while the metal-poor stars show higher vertical velocities, which is compatible with the more random orbits of the halo stars. The variation of Galactic velocity components in the chemical parameter space suggests a transition between the disk and halo stars around [M/H]$\approxeq$$-$0.5 dex.

The color maps in both Figures 40 and 41 emphasize the distinction between metal-poor halo stars and metal-rich disk stars, but the details of certain correlations between metallicity and velocity for the disk stars are mostly missed in these diagrams. For this reason, we plot the velocity components U, V, and W versus [M/H] and [M/H]+[$\alpha$/Fe] for the above-defined combined  Groups UV+UW, UV+VW, and UW+VW, respectively, but with narrower ranges of velocities for each group:  $-$200$\leq$U$\leq$200 km/s (Group UV+UW), $-$40$\leq$V$\leq$260 km/s (Group UV+VW) , and $-$100$\leq$W$\leq$100 km/s (Group UW+VW),  using the best-fit chemical parameter estimates inferred from the constant-gravity approach, as shown in Figure 42. The chemical parameter values are again randomized within the respective step size of the model grid. The plots related to the “reduced-correlation method” do not convey independent information and are not contained in this paper.  Apart from the offset  in the V values by $\sim$220 km/s due to our converted velocities to  the Galactic rest frame, the distributions are quite similar to the metallicity-velocity plots of   N04 (Figure 32 in that paper) for their subset of  14000 nearby F and G dwarfs, and the distribution of our M dwarfs/subdwarfs  shows the same chemodynamical substructure.

We identify three stellar streams in the bottom middle panel (“e”) of Figure 42, as  zoomed in Figure 43. There is a group of stars at the top of the distribution (V$\sim$220 km/s) which has an average value of  [M/H]+[$\alpha$/Fe]$\sim$0.0 dex and appears slightly shifted to the left on the plot relative to the overall distribution  of the disk stars.  These stars are  associated with the “Sirius Stream”  that belongs to  the outer disk and are expected to be  slightly more metal-poor than the average disk stars. This group can be compared to the bulk of the disk stars (V$\sim$180-190 km/s) that have  an average value of  [M/H]+[$\alpha$/Fe]$\sim$0.1-0.2 dex and are marginally  more metal-rich than the stars in the Sirius Stream. Two other streams, i.e., Hyades and Hercules streams, are also marked in Figure 43. Additional substructure may originate from moving groups, or as mentioned in N04, from inhomogeneous Galactic potentials, e.g., by stochastic spiral waves (e.g., De Simone et al. 2004) or the bar (e.g., Fux 2001).

In general, the range of metallicities for the disk stars ($\sim$0.5-0.6 dex) is remarkably similar to that of N04, which reinforces that our model-fit parameter values are highly precise. Furthermore, the average metallicity of our disk stars  has only an offset $\sim$0.1-0.2 dex relative to the average value of N04 (while the chemical parameter values from the variable-gravity approach  show a larger difference in average metallicity $\sim$0.3-0.35 dex with respect to that of N04), which indicates that our parameter estimates using the constant-gravity approach are accurate at the level of  $\sim$0.1-0.2 dex. Although the photometric surface gravities are relatively approximate and perhaps not entirely reliable (notably for unresolved binary stars), these values still yield reasonably accurate  estimates of T$_\textrm{\footnotesize{eff}}$, [M/H] and [M/H]+[$\alpha$/Fe]. If these estimates had very large uncertainties, we would not find subtle features in abundance-velocity diagrams and we would not reach  such a consistency with other previous studies.

lcccccc [h!]

Spectroscopic Catalog of the 3745 stars: Astrometry and Kinematics

Object        R.A. (Deg)      Decl (Deg)
$\mu_\textrm{\scriptsize{R.A.}}$ ($\arcsec$$\textrm{yr}^{-1}$)      $\mu_\textrm{\scriptsize{Decl}}$ ($\arcsec$$\textrm{yr}^{-1}$)     prlx ($\arcsec$)      Gaia Flag
PM J00012+0659      0.31397      6.99282      $-$0.4363      $-$0.0834      0.04278      DR3

PM J00031+0616      0.78061      6.27530        0.2389      $-$0.5128      0.03430      DR3

PM J00051+4547      1.30092     45.78589        0.8707      $-$0.1514      0.08693      DR3

PM J00077+6022      1.93035     60.38166        0.3216      $-$0.0031      0.06500      DR3

PM J00078+6736      1.96105      67.60664      $-$0.0559      $-$0.0911      0.03965      DR3

PM J00081+4757      2.02595      47.95068      $-$0.1241        0.0044      0.05362     DR3

PM J00088+2050      2.22444      20.83929      $-$0.0486      $-$0.2602      0.05526      DR2

PM J00094+6549      2.35085      65.81710      $-$0.0827      $-$0.0453      0.00977      DR3

PM J00095+6548      2.38440      65.80898      $-$0.0788      $-$0.0450      0.00963      DR3

PM J00101+1327      2.54048      13.45336      $-$0.2956      $-$0.4169      0.02281      DR3

PM J00110+0420      2.75393      4.33799        0.1485      $-$0.5234      0.01229      DR3

PM J00119+3303      2.98221      33.05296      $-$0.5566      $-$0.3965      0.04775      DR3

PM J00120+0206      3.01249      2.10687        0.3022        0.0658      0.04209      DR3

PM J00122+3028      3.05619      30.47884        0.0584       $-$0.0231       0.01166      DR2

PM J00123+6142      3.07600      61.71170      $-$0.0764        0.0379      0.02986      DR3

PM J00131+7023      3.29868      70.39858        0.0432        0.1392     0.03153      DR3

PM J00133+3908      3.33305     39.14572      $-$0.4503      $-$0.1882      0.02713      DR3

PM J00137+8039      3.43557      80.66449        0.2508        0.1944      0.05203      DR2

PM J00138+3537      3.45507      35.61685      $-$0.1917      $-$0.4274      0.03687      DR2

PM J00144+3609N      3.60644      36.15699        0.1998      $-$0.0905      0.00607      DR3

This table is available in its entirety in a machine-readable  form in the online journal. A portion is shown here for guidance regarding its form and content.

lcccccccc [h!]
Spectroscopic Catalog of the 3745 stars: Classification and Photometry

Object     Spectral Type       Metallicity Class
G         G$_\textrm{\footnotesize{BP}}$      G$_\textrm{\footnotesize{RP}}$       J     H     K
PM J00012+0659     dM6.0     2     14.67     16.77     13.33     11.29     10.74     10.42

PM J00031+0616     dM4.5     2     14.00     15.67     12.76     11.04     10.53     10.30

PM J00051+4547     dM1.5     2     9.09     10.18     8.05     6.70     6.10     5.85

PM J00077+6022     dM5.0     3     12.40     13.64     10.70     8.91     8.33     8.05

PM J00078+6736     dM2.0     2     11.03     12.12     9.81     8.36     7.73     7.51

PM J00081+4757     dM4.0     2     11.42     12.99     10.20     8.52     8.00     7.68

PM J00088+2050     dM5.0     2     11.99     13.77     10.72     8.87     8.26     8.01

PM J00094+6549     dM0.5     3     12.66     13.56     11.71     10.50     9.85     9.67

PM J00095+6548     dM3.5     2     14.79     16.13     13.65     12.14     11.56     11.33

PM J00101+1327     dM4.5     3     14.92     16.44     13.72     12.12     11.59     11.34

PM J00110+0420     esdM6.0     8     16.89     18.14     15.79     14.34     13.81     13.76

PM J00119+3303     dM4.0     2     11.83     13.27     10.65     9.07     8.40     8.16

PM J00120+0206     dM4.5     2     13.69     15.45     12.43     10.59     10.01     9.72

PM J00122+3028     dM6.5     2     13.39     15.10     12.11     10.24     9.68     9.41

PM J00123+6142     dM4.5     2     13.17     14.70     11.93     9.97     9.30     8.95

PM J00131+7023     dM1.5     3     10.44     11.40     9.47     8.26     7.59     7.39

PM J00133+3908     dM3.0     3     12.66     13.86     11.58     10.18     9.61     9.36

PM J00137+8039     dM6.5     3     14.42     15.65     12.93     10.94     10.37     10.06

PM J00138+3537     dM6.5     2     13.69     15.36     12.45     10.68     10.12     9.86

PM J00144+3609N     dM3.0     3     15.93     17.14     14.84     13.45     12.93     12.71

This table is available in its entirety in a machine-readable  form in the online journal. A portion is shown here for guidance regarding its form and content.

Classification and Chemical Parameters of the 74 Common Proper-Motion Pairs

5c  |Companion       5cPrimary

Name     Sp-Type     MC     [M/H]     [$\alpha$/Fe]     Name     Sp-Type     MC     [M/H]     [$\alpha$/Fe]

PM J00144+3609S    dM4.0    3    $-$0.25    +0.175    PM J00144+3609N    dM3.0    3    $-$0.10    +0.075

PM J00162+1951W    dM4.5    2    +0.15    +0.200    PM J00162+1951E    dM4.0    1    +0.10    +0.250

PM J00277+0936N    dM4.0    2    +0.00    +0.125    PM J00277+0936S    dM4.0    2    +0.15    +0.100

PM J00358+5241N    dM4.5    2    $-$0.10    +0.175    PM J00358+5241S    dM3.0    2    $-$0.10    +0.100

PM J00512+5840    dM4.5    2    $-$0.05    +0.150    PM J00512+5839    dM3.5    2    $-$0.05    +0.125

PM J01024+4101    dM4.5    2    $-$0.15    +0.100    PM J01024+4102    dM3.0    3    +0.05    $-$0.025

PM J01544+5741N    dM4.0    2    +0.05    +0.075    PM J01544+5741S    dM4.0    2    +0.00    +0.050

PM J02325+5313    dM5.0    2    $-$0.20    +0.225    PM J02324+5313    dM3.0    2    $-$0.25    +0.175

PM J02340+0950W    dM6.5    3    $-$0.10    +0.150    PM J02340+0950E    dM3.0    3    $-$0.40    +0.300

PM J02498+3945E    dM3.5    2    $-$0.15    +0.125    PM J02498+3945W    dM3.5    2    $-$0.10    +0.150

PM J02515+5922E    dM4.0    2    $-$0.15    +0.175    PM J02515+5922W    dM4.0    2    $-$0.10    +0.200

PM J03055+1149S    dM3.5    3    +0.00    +0.050    PM J03055+1149N    dM3.0    3    +0.10    +0.000

PM J03101$-$0046N    dM3.5    3    $-$0.20    +0.225    PM J03101$-$0046S    dM3.5    2    $-$0.35    +0.300

PM J03265+5908W    dM4.5    2    $-$0.10    +0.100    PM J03265+5908E    dM4.0    3    $-$0.05    +0.075

PM J03375+1751S    dM4.0    2    +0.10    +0.025    PM J03375+1751N    dM3.0    3    +0.00    +0.000

PM J03576+5715E    sdM5.0    5    $-$0.70    +0.200    PM J03576+5715W    sdM4.0    6    $-$0.65    +0.150

PM J04059+7116W    dM6.0    2    $-$0.05    +0.225    PM J04059+7116E    dM4.5    2    +0.05    +0.175

PM J04248+0106E    dM5.0    2    +0.05    +0.150    PM J04248+0106W    dM3.0    2    $-$0.15    +0.150

PM J04278+2630S    dM4.5    2    $-$0.05    +0.200    PM J04278+2630N    dM3.5    2    $-$0.05    +0.125

PM J05342+1019S    dM4.0    2    +0.05    +0.150    PM J05342+1019N    dM3.5    2    $-$0.05    +0.125

PM J05422$-$2219    dM6.5    3    $-$0.30    +0.250    PM J05422$-$2220    dM4.0    3    $-$0.35    +0.225

PM J06007+6808    dM4.0    2    $-$0.05    +0.150    PM J06008+6809    dM4.0    2    $-$0.10    +0.150

PM J06381+2219E    dM4.0    2    $-$0.20    +0.225    PM J06381+2219W    dM3.0    2    $-$0.05    +0.100

PM J07390+7734W    dM5.5    2    +0.05    +0.175    PM J07390+7734E    dM4.5    2    +0.00    +0.150

PM J07430+5109E    dM3.5    3    +0.05    0.075    PM J07430+5109W    dM3.0    2    +0.00    +0.050

PM J08398+0856E    dM4.5    3    $-$0.15    +0.075    PM J08398+0856W    sdM3.0    4    $-$0.05    +0.025

PM J09070+7226    dM5.5    2    +0.00    +0.200    PM J09069+7226    dM4.5    2    +0.00    +0.175

PM J09112+0127N    dM4.5    2    +0.25    +0.100    PM J09112+0127S    dM4.0    1    +0.15    +0.125

PM J09593+4350E    dM5.0    3    $-$0.20    +0.125    PM J09593+4350W    dM4.5    3    $-$0.10    +0.075

PM J10182$-$2028W    dM6.5    2    +0.25    +0.250    PM J10182$-$2028E    dM5.5    2    +0.20    +0.225

PM J10260+5027N    dM4.0    2    +0.00    +0.100    PM J10260+5027S    dM4.0    2    +0.00    +0.125

PM J10344+4618    dM5.5    2    +0.15    +0.125    PM J10345+4618    dM3.5    2    +0.00    +0.100

PM J10449+3224    dM5.0    2    +0.05    +0.125    PM J10448+3224    dM3.5    2    $-$0.10    +0.100

PM J11285+5643N    dM4.5    2    $-$0.05    +0.125    PM J11285+5643S    dM4.5    2    +0.00    +0.125

PM J11488+1800E    dM4.0    2    +0.05    +0.100    PM J11488+1800W    dM4.0    2    +0.00    +0.100

PM J12091+4735    dM5.0    2    $-$0.05    +0.150    PM J12091+4736    dM5.0    2    $-$0.05    +0.125

PM J14282+6349    dM4.0    3    $-$0.15    +0.075    PM J14283+6349    dM4.0    3    $-$0.20    +0.100

5c  |Companion       5c Primary

Name     Sp-Type     MC     [M/H]     [$\alpha$/Fe]     Name     Sp-Type     MC     [M/H]     [$\alpha$/Fe]

PM J14372+7536    sdM3.0    4    $-$0.35    +0.075    PM J14371+7536    dM3.0    3    $-$0.40    +0.175

PM J14518+5147W    dM5.0    3    $-$0.55    +0.275    PM J14518+5147E    dM3.0    3    $-$0.40    +0.175

PM J14567+1245    dM4.0    3    +0.20    $-$0.025    PM J14568+1245    dM4.0    3    +0.20    $-$0.025

PM J15091+1513E    dM6.5    2    +0.00    +0.200    PM J15091+1513W    dM3.0    2    +0.15    +0.025

PM J15308+5608N    esdM6.0    7    $-$1.30    +0.275    PM J15308+5608S    esdM6.0    7    $-$1.25    +0.250

PM J15353+1743    dM5.0    2    $-$0.30    +0.225    PM J15353+1742    dM3.5    3    $-$0.15    +0.150

PM J15378+0017E    sdM5.0    5    $-$1.05    +0.225    PM J15378+0017W    esdM4    7    $-$1.20    +0.325

PM J17023+1647N    dM3.5    3    $-$0.30    +0.200    PM J17023+1647S    dM3.5    3    $-$0.10    +0.050

PM J17184$-$2246    dM5.0    2    +0.00    +0.125    PM J17184$-$2245    dM4.0    3    $-$0.15    +0.200

PM J17193$-$2949N    dM5.0    2    $-$0.05    +0.175    PM J17193$-$2949S    dM4.5    2    +0.00    +0.150

PM J17277+5200N    dM5.0    2    +0.05    +0.100    PM J17277+5200S    dM5.0    2    +0.05    +0.100

PM J18180+3846W    dM4.5    2    $-$0.25    +0.275    PM J18180+3846E    dM3.5    2    $-$0.25    +0.225

PM J18216+3840S    dM5.0    2    +0.05    +0.100    PM J18216+3840N    dM4.0    3    $-$0.05    +0.075

PM J18427+5937S    dM4.0    2    $-$0.20    +0.275    PM J18427+5937N    dM3.5    2    $-$0.25    +0.300

PM J18447+3238    dM5.5    3    +0.20    +0.000    PM J18447+3237    dM3.0    2    +0.00    +0.050

PM J19096+1401E    dM3.5    2    +0.00    +0.075    PM J19096+1401W    dM3.0    2    +0.05    +0.025

PM J19113+5224E    dM3.5    2    +0.00    +0.100    PM J19113+5224W    dM3.5    1    +0.00    +0.125

PM J19312+3607    dM5.0    2    +0.15    +0.200    PM J19311+3608    dM4.5    2    +0.10    +0.200

PM J19317+4506E    dM4.0    2    +0.15    +0.000    PM J19317+4506W    dM4.0    2    +0.15    +0.025

PM J19388+3512E    dM5.0    3    $-$0.35    +0.200    PM J19388+3512W    dM4.0    3    +0.00    +0.000

PM J19433+5433E    dM4.0    2    +0.35    +0.000    PM J19433+5433W    dM4.0    3    +0.30    +0.000

PM J20231+1844E    dM4.0    3    $-$0.30    +0.200    PM J20231+1844W    dM3.5    2    $-$0.35    +0.275

PM J21013+3314    dM4.0    3    $-$0.15    +0.100    PM J21012+3314    dM3.5    3    $-$0.05    +0.050

PM J21080+3116S    dM5.5    2    $-$0.20    +0.200    PM J21080+3116N    dM3.0    2    $-$0.15    +0.150

PM J21160+2951E    dM4.0    2    $-$0.10    +0.050    PM J21160+2951W    dM4.0    2    $-$0.15    +0.100

PM J22173$-$0848S    dM5.0    2    $-$0.20    +0.200    PM J22173-0848N    dM5.0    3    $-$0.20    +0.175

PM J22176+6010W    dM4.0    2    +0.00    +0.125    PM J22176+6010E    dM3.5    2    +0.00    +0.100

PM J22297+2226N    dM6.5    2    +0.20    +0.050    PM J22297+2226S    dM3.0    3    $-$0.20    +0.125

PM J22343+3129    dM4.0    2    $-$0.10    +0.175    PM J22342+3128    dM3.5    2    +0.05    +0.075

PM J22412$-$1513E    dM3.5    1    $-$0.05    +0.150    PM J22412$-$1513W    dM3.5    1    $-$0.05    +0.225

PM J23257+1735    dM6.5    2    $-$0.35    +0.225    PM J23258+1735    dM4.5    3    $-$0.30    +0.200

PM J23293+4127    dM4.5    2    $-$0.05    +0.100    PM J23293+4128    dM4.0    2    +0.05    +0.050

PM J23310+0842N    dM4.0    2    $-$0.10    +0.100    PM J23310+0842S    dM3.0    2    $-$0.15    +0.150

PM J23578+7836    dM3.5    3    $-$0.10    +0.000    PM J23580+7836    dM3.0    3    $-$0.10    +0.025

PM J02340+0950W    dM6.5    3    $-$0.10    +0.150    PM J02340+0950E    dM3.0    3    $-$0.40    +0.300

PM J10182$-$2028W    dM6.5    2    +0.25    +0.250    PM J10182$-$2028E    dM5.5    2    +0.20    +0.225

PM J17184$-$2246    dM5.0    2    +0.00    +0.125    PM J17184$-$2245    dM4.0    3    $-$0.15    +0.200

Spectroscopic Catalog of the 3745 stars: Model-Fit Stellar Parameters

This table is available in its entirety in a machine-readable  form in the online journal. A portion is shown here for guidance regarding its form and content.

Object     ($\textrm{T}_\textrm{eff}$)$_\textrm{N}$ (K)     $\sigma$($\textrm{T}_\textrm{eff}$)$_\textrm{N}$ (K)       [M/H]$_\textrm{N}$        $\sigma$[M/H]$_\textrm{N}$     [$\alpha$/Fe]$_\textrm{N}$     $\sigma$[$\alpha$/Fe]$_\textrm{N}$      ($\textrm{T}_\textrm{eff}$)$_\textrm{RC}$ (K)       $\sigma$($\textrm{T}_\textrm{eff}$)$_\textrm{RC}$ (K)       [M/H]$_\textrm{RC}$     $\sigma$[M/H]$_\textrm{RC}$     [$\alpha$/Fe]$_\textrm{RC}$     $\sigma$[$\alpha$/Fe]$_\textrm{RC}$     log(g)$_\textrm{Phot}$

PM J00012+0659     2950.0     25.1     $-$0.15     0.18     +0.200     0.102     2950.0      0.0     $-$0.18     0.12     +0.215     0.049     5.12

PM J00031+0616     3200.0     33.5     +0.00     0.18     +0.150     0.097     3191.7     20.4     $-$0.04     0.14     +0.150     0.089     5.03

PM J00051+4547     3850.0     132.9     $-$0.10     0.18     +0.075     0.085     3816.7     66.1     $-$0.12     0.08     +0.100     0.055     4.72

PM J00077+6022     3150.0     37.2     $-$0.30     0.18     +0.125     0.097     3150.0      0.0     $-$0.30     0.07     +0.113     0.018     4.90

PM J00078+6736     3700.0     127.5     $-$0.05     0.16     +0.125     0.093     3700.0       0.0     $-$0.08     0.04     +0.138     0.038     4.69

PM J00081+4757     3300.0     87.2     $-$0.25     0.19     +0.200     0.100     3300.0      0.0     $-$0.27     0.06     +0.214     0.028     4.82

PM J00088+2050     3100.0     24.8     $-$0.10     0.18     +0.125     0.093     3100.0      0.0     $-$0.09     0.05     +0.119     0.039     4.87

PM J00094+6549     4000.0     81.1     +0.00     0.20     +0.075     0.081     4000.0      0.0     $-$0.12     0.13     +0.167     0.052     4.58

PM J00095+6548     3400.0     86.1     $-$0.10     0.17     +0.150     0.093     3433.3     57.7     $-$0.07     0.10     +0.150     0.000     4.81

PM J00101+1327     3250.0     32.9     $-$0.40     0.17     +0.200     0.101     3250.0      0.0     $-$0.42     0.08     +0.222     0.042     5.06

PM J00110+0420     3300.0     37.6     $-$1.85     0.25     +0.400     0.119     3307.1     18.9     $-$1.85     0.19     +0.396     0.051     5.22

PM J00119+3303     3300.0     75.3     +0.05     0.16     +0.050     0.078     3300.0      0.0     +0.04     0.07     +0.050     0.038    4.85

PM J00120+0206     3200.0     38.5     +0.20     0.17     +0.050     0.081     3188.9     22.1     +0.17     0.07     +0.056     0.037     5.01

PM J00122+3028     3100.0     68.1     +0.00     0.21     +0.100     0.095     3100.0      0.0     +0.00     0.00     +0.100     0.000     4.59

PM J00123+6142     3300.0     92.9     +0.00     0.17     +0.225     0.094     3316.7     35.4     $-$0.01     0.07     +0.236     0.052     4.82

PM J00131+7023     3900.0     93.1     +0.25     0.21     $-$0.025     0.069     3900.0      0.0     +0.25     0.04     $-$0.025     0.000     4.61

PM J00133+3908     3500.0     104.5     $-$0.05     0.20     +0.000     0.075     3500      0.0     $-$0.14     0.18     +0.025     0.100     4.85

PM J00137+8039     3300.0     99.3     +0.50     0.09     +0.125     0.083     3327.8     83.3     +0.47     0.07     +0.142     0.076     5.13

PM J00138+3537     3200.0     37.7     +0.00     0.18     +0.100     0.093     3200.0      0.0     +0.08     0.04     +0.063     0.018     5.00

PM J00144+3609N     3500.0     95.5     $-$0.10     0.17     +0.075     0.087     3533.3     35.4     $-$0.11     0.15     +0.108     0.082     4.86

§ SUMMARY AND CONCLUSION

We present a synthetic  fitting pipeline that can be applied to low-resolution spectra of M dwarfs/subdwarfs to infer their physical parameters. Since such spectra are available in large numbers, their derived chemical parameter values  can provide excellent clues on the Galactic chemodynamical  evolution. Moreover,  the atmosphere of these low-mass stars has remained in a near-pristine chemical state since they formed and their abundance properties  can be therefore used as reliable  probes of   the Galaxy's early chemical enrichment. However, we describe some complications  in our low-resolution model-fit pipeline that have to be taken into account when interpreting the results, as summarized below:

* As can be seen from Figures 22-25,  the stars generally resist to be matched with high-temperature, high-metallicity models (T$_\textrm{\footnotesize{eff}}$>3550 K and [M/H]>+0.1 dex), which is most likely due to issues in high-temperature synthetic models. Our high-temperature stars are systematically fitted with metallicity values within a specific  range (i.e., $-$0.5$\leq$[M/H]$\leq$+0.1), and are thus excluded from the present analysis.

* As presented in Figures 24, 25, 27, 28, and the bottom panels of Figures 29 and 30, compared to the results using the constant (photometric) surface gravities, when log g becomes a free parameter, due to the strong degeneracy between this parameter and [M/H], a large number of stars with near-solar metallicities move towards higher values (or even higher ends) of log g and [M/H]. This shows that the variable-gravity approach  may cause  large systematic uncertainties in the inferred parameter values.

* As shown in Figure 27 and the bottom panels of Figures 29 and 30, in comparison with the results using the constant (photometric) surface gravities, when log g is allowed to vary, a significant number of stars with near-solar metallicities shift towards higher values of log g and  [$\alpha$/Fe], which is due to the  degeneracy effect caused by these two parameters. This again indicates the unreliability of the derived parameter values from the variable-gravity approach.

* For either gravity-modeling approach, there is a systematic trend between T$_\textrm{\footnotesize{eff}}$ and [$\alpha$/Fe]. As it can clearly be seen from the middle panels of Figures 29 and 30, the higher-temperature stars tend to be fitted with the models having  lower values of [$\alpha$/Fe] (that are less affected by molecular bands) and vice versa, which is likely owing to deficiencies in the model spectra.

* The tight correlation between the resulting values of  T$_\textrm{\footnotesize{eff}}$  and log g inferred from the constant-gravity approach  appears as a relation between log g and [$\alpha$/Fe], which  can be perceived from the panels “c” and “e” in Figures 29 and 30. When  log g is kept fixed,  the derived values of [$\alpha$/Fe] follow nearly the same trend with log g (panel “e”) as with T$_\textrm{\footnotesize{eff}}$ (panel “c”). Since log g  is constant, the degeneracy between log g and [$\alpha$/Fe] plays no role in the relation between these two parameters.

* For either gravity-modeling approach, as shown in the bottom panels of Figures 29 and 30,  the tilted distributions of stars with nearly the same values of log g is a reflection of the tight anti-correlation between [M/H] and [$\alpha$/Fe].

It should be noted that all these complications and subtleties  are independent of the selected set of initial values, the formulation of  $\chi$$^\textrm{\footnotesize{2}}$  (i.e., whether including the observational flux error or not), and the extent of the grids over which the minimization routines are performed.  Limitations  in the model atmospheres to correctly address the effect of molecular bands  caused by T$_\textrm{\footnotesize{eff}}$ and the two chemical parameters, i.e., [M/H] and  [$\alpha$/Fe], have raised some systematic trends in the distributions of our resulting parameter values. Although the exclusion  of high-temperature stars (T$_\textrm{\footnotesize{eff}}$ >3550 K) reduces  these systematics to a great extent, there is still a trend between T$_\textrm{\footnotesize{eff}}$ and [$\alpha$/Fe], even using the constant-gravity approach,  which is unlikely  due to  sample selection effects or problems in the model-fit pipeline. Despite this systematic relation, we find the best-fit values of  T$_\textrm{\footnotesize{eff}}$, [M/H], and [M/H]+[$\alpha$/Fe] highly precise. This precision is confirmed by

* The clear stratification of T$_\textrm{\footnotesize{eff}}$ in the HR diagram (Figures 22 and 23),

* The clear stratification of [M/H] and [M/H]+[$\alpha$/Fe] in the HR diagram (Figures 24-26),

* The similarity between the distribution of our stars (left panels in Figures 29 and 30)  and that of  other studies (e.g., GALAH DR3,  Buder et al. 2021, Figure 5) in the abundance diagram of [$\alpha$/Fe] versus [M/H]. Regardless of  some systematic offsets in parameter values between the two studies, which is due to the difference in the spectral resolutions, methods and model atmospheres, both distributions show nearly the same trend between [M/H] and  [$\alpha$/Fe].

* Chemical homogeneity between the components of a set of binary systems (top panels in Figure 38),

* Revealing some substructure like stellar “streams” in abundance-velocity diagrams (Figure 43),

* The notable similarity between the distributions of our stars (top panels in Figure 42) and those of other studies (e.g., N04, Figure 32) in  metallicity-velocity diagrams.

While the photometric surface gravities may not be accurate, these values still yield reasonable results when being kept fixed  in the synthetic fitting pipeline. It should be recalled that the relation between these gravity values and the temperatures derived from the constant-gravity approach is predicted by theoretical isochrones. In addition, the evident contrast in surface gravity between metal-poor M subdwarfs and their metal-rich counterparts is consistent with the relative size of these stars (Section 6.3). To achieve more accurate stellar parameters from low-resolution spectroscopy, more reliable values of surface gravity from future extensive studies will thus be required.

Table 4 lists the best-fit  values of T$_\textrm{\footnotesize{eff}}$, [M/H], and [$\alpha$/Fe] inferred from both the normal (with a subscript “N”) and the reduced-correlation (with a subscript “RC”) methods using the constant-gravity approach for the 3745 stars. $\sigma$X$_\textrm{\footnotesize{N}}$ where X is T$_\textrm{\footnotesize{eff}}$, [M/H], or [$\alpha$/Fe] stands for the corresponding parameter  error  that is calculated based on the method described in Section 5. $\sigma$X$_\textrm{\footnotesize{RC}}$ denotes the standard deviation of the ten measurements from the mean value for each parameter T$_\textrm{\footnotesize{eff}}$, [M/H], and [$\alpha$/Fe] (Sections 4 and 5). The photometric surface gravities are also listed in this table.  It is important to note that all effective temperatures are included in the table, however, caution should be taken when using high temperature stars (T$_\textrm{\footnotesize{eff}}$>3550 K) whose parameter values most likely suffer from large uncertainties.

In our future work, we aim to expand our spectroscopic sample using a large SDSS/SEGUE catalog. We will apply the pipeline to those spectra with good quality and examine their distribution in some important photometric, chemical and kinematic diagrams. The larger number of metal-poor M subdwarfs in the SDSS/SEGUE sample may disclose some substructure in the Galactic halo.

We greatly thank the anonymous referee for their careful reading and helpful comments and  suggestions that significantly improved our manuscript. We would also like to acknowledge Ian Czekala for his useful discussions and suggestions. We extend our thanks to Douglas Gies, Russel White, Philip Muirhead, and Derek Homeier for their extremely insightful comments and suggestions. We would like to appreciate Ilija Medan for his help with the Gaia data. We finally wish to thank Justin Cantrell and Jeremy Simmons for their technical support. This material is based on work supported by the National Science Foundation under grant No. AST 09-08419, the National Aeronautics and Space Administration under grant Nos. NNX15AV65G, NNX16AI63G, and NNX16AI62G issued through the SMD/Astrophysics Division as part of the K2 Guest Observer Program. Parts of this research were supported  by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

§ REFERENCES

Adibekyan, V. Z., Figueira, P., Santos, N. C., et al. 2013, A&A, 554, A44 (DOI: 10.1051/0004-6361/201321520)

Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 (DOI: 10.1051/0004-6361/201219401)

Allard, F., & Hauschildt, P. H. 1995, ApJ, 445, 433 (DOI: 10.1086/175708)

An, D. 2019. ApJL, 878, L31 (DOI: 10.3847/2041-8213/ab2467)

Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42 (DOI: 10.1051/0004-6361/201425481)

Belokurov, V.,  Penoyre, Z.,  Oh, S. et al. 2020, MNRAS, 496, 1922 (DOI: 10.1093/mnras/staa1522)

Bergemann, M., & Nordlander, T. 2014, Determination of Atmospheric Parameters of B-, A-, F- and G-Type (Basel: Springer), 169 (DOI: 10.1007/978-3-319-06956-2$\_$16)

Bessell, M. S. 1991, AJ, 101, 662 (DOI: 10.1086/115714)

Bochanski, J. J., Hawley, S. L., Covey, K. R., et al. 2010, AJ, 139, 2679 DOI: 10.1088/0004-6256/139/6/2679

Buder, S., Sharma, S., Kos, J., et al. 2021, MNRAS, 506, 150 (DOI: 10.1093/mnras/stab1242)

Chavez, J., & Lambert, D. L. 2009, ApJ, 699, 1906 (DOI: 10.1088/0004-637X/699/2/1906)

Chiappini, C. 2001, AmSci, 89, 506 (DOI: 10.1511/2001.6.506)

Chiappini, C., Matteucci, F., Beers, T., & Nomoto, K. 1999, ApJ, 515, 226 (DOI: 10.1086/307006)

Chiappini, C., Matteucci, F., & Romano, D. 2001, ApJ, 554, 1044 (DOI: 10.1086/321427)

Choi, J.,  Dotter, A.,  Conroy, C., et al. 2016, ApJ, 823, 102 (DOI: 10.3847/0004-637X/823/2/102)

Croswell, K. 1995, The Alchemy of the Heavens: Searching for Meaning in the Milky Way (New York: Anchor Books)

Czekala, I., Andrews, S. M., Mandel, K. S., et al.  2015, ApJ, 812, 128 (DOI: 10.1088/0004-637X/812/2/128)

De Simone, R. A., Wu, X., & Tremaine, S. 2004, MNRAS, 350, 627 (DOI:  10.1111/j.1365-2966.2004.07675.x)

Dotter, A., Chaboyer, B., Jevremovic, D. et al. 2008, ApJS, 178, 89, (DOI: 10.1086/589654)

Dotter, A. 2016, ApJS, 222, 8 (DOI:10.3847/0067-0049/222/1/8)

Du, C-H., Zhou, X., Ma, J., et al. 2004, AJ, 128, 2265 (DOI: 10.1086/424858)

Fux, R. 2001, A&A, 373, 511 (DOI: 10.1051/0004-6361:20010561)

Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1  (DOI: 10.1051/0004-6361/201833051)

Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., et al. 2021, A&A, 649, A1 (DOI: 10.1051/0004-6361/202039657)

Gao, X., Lind, K., Amarsi, A. M., et al. 2018, MNRAS, 481, 2666 (DOI: 10.1093/mnras/sty2414)

Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951 (DOI: 10.1051/0004-6361:200809724)

Hawkins, K.,  Lucey, M.,  Ting, Y.-S., et al. 2020, MNRAS, 492, 1164  (DOI: 10.1093/mnras/stz3132)

Hawkins, K.,  Jofré, P.,  Masseron, T., &   Gilmore, G. 2015, MNRAS, 2015, 453, 758 (DOI (10.1093/mnras/stv1586)

Hayden, M. R., Bovy, J., Holtzman, J. A., et al. 2015, ApJ, 808, 132 (DOI: 10.1088/0004-637X/808/2/132)

Haywood, M. 2001, MNRAS, 325, 1365 (DOI: 10.1046/j.1365-8711.2001.04510.x)

Hejazi, N., De Robertis, M. M., & Dawson, P. C. 2015, AJ, 149, 140 (DOI: 10.1088/0004-6256/149/4/140)

Hejazi, N., Lepine, S., Homeier, D., Rich, R. M., & Shara, M. M. 2020, AJ,  159, 30 (DOI: 10.3847/1538-3881/ab563c)

Heringer, E., Pritchet, C., & van Kerkwijk, M. H. 2019, ApJ, 882, 52 (DOI: 10.3847/1538-4357/ab32dd)

Hinkle, K. H.,  Wallace, L., &  Livingston, W. 2003. AAS, Meeting No. 203, 35,1260

Kervella, P.,  Arenou, F., &  Thevenin, F. 2022, A&A, 657, A7 (DOI: 10.1051/0004-6361/202142146)

Kesseli, A. Y., Kirkpatrick, J. D., Sergio, B. F., et al. 2019, AJ, 157, 63 (DOI: 10.3847/1538-3881/aae982)

Kippenhahn, R., & Weigert, A. 1990, Stellar Structure and Evolution (Springer-Verlag Berlin Heidelberg New York)

Kouwenhoven, M. B. N., Goodwin, S. P., Parker, R. J., et al.  2010, MNRAS, 404, 1835 (DOI: 10.1111/j.1365-2966.2010.16399.x)

Kuznetsov, M. K., del Burgo, C., Pavlenko, Ya. V., & Frith, J. 2019, ApJ, 878, 134 (DOI: 10.3847/1538-4357/ab1fe9)

Lee, Y. S., Beers, T. C, Prieto, C. A., et al. 2011a, AJ,141, 90  (DOI: 10.1088/0004-6256/141/3/90)

Lee, Y. S., Beers, T. C., An, D., et al. 2011b, ApJ, 738, 187 (DOI: 10.1088/0004-637X/738/2/187)

Lee, J.-E., Lee, S., Dunham, M. M., et al. 2017, NatAs, 1, 0172  (DOI: 10.1038/s41550-017-0172)

Lépine, S., & Shara, M. M. 2005, AJ, 129, 1483 (DOI: 10.1086/427854)

Lépine, S. 2005, AJ, 130, 1680 (DOI:  10.1086/432792)

Lépine, S., Rich, R. M., & Shara, M. M. 2007, ApJ, 669, 1235 (DOI: 10.1086/521614)

Lépine, S., Hilton, E. J., Mann, A. W., et al. 2013, AJ, 145, 102 (DOI: 10.1088/0004-6256/145/4/102)

Mann, A. W., Brewer, J. M., Gaidos, E., Lépine, S., & Hilton, E. J. 2013, AJ, 145, 52 (DOI: 10.1088/0004-6256/145/2/52)

Mann, A. W.,  Feiden, G. A., Gaidos, E., et al. 2015, ApJ, 804, 64 (DOI: 10.1088/0004-637X/804/1/64)

Mann, A. W.,  Dupuy, T., Kraus, A. L., et al. 2019, ApJ, 871, 63  (DOI: 10.3847/1538-4357/aaf3bc)

Moeckel, N. & Clarke C. J.  2011, MNRAS, 415, 1179  (DOI: 10.1111/j.1365-2966.2011.18731.x)

Nordlander, T., Amarsi, A. M., Lind, K., et al. 2017, A&A, 597, A6

Nordström, B.,  Mayor, M.,  Andersen, J., et al. 2004,  A&A, 418, 989 (DOI: 10.1051/0004-6361:20035959)

Offner, S. S. R., Kratter, K. M., Matzner, C. D., et al.  2010, ApJ, 725, 1485  (DOI: 10.1088/0004-637X/725/2/1485)

Pagel, B. E. 1997, Nucleosynthesis and Chemical Evolution of Galaxies (Cambridge: Cambridge Univ. Press)

Passegger, V. M., Wende-von Berg, S., & Reiners, A. 2016, A&A, 587, A19  (DOI: 10.1051/0004-6361/201322261)

Passegger, V. M., Reiners, A., Jeffers, S. V., et al.  2018, A&A, 615, A6 (DOI: 10.1051/0004-6361/201732312)

Pipino, A., &  Matteucci, F. 2009, The Ages of Stars, Proceedings of the International Astronomical Union, IAU Symposium,  258, 39 (DOI: 10.1017/S174392130903169X)

Prsa, A,  Harmanec, P.,  Torres, G., et al. 2016, AJ, 152, 41 (DOI: 10.3847/0004-6256/152/2/41)

Rains, A. D., Zerjal, M., Ireland, M. J., et al. 2021, MNRAS, 504, 5788 (DOI: 10.1093/mnras/stab1167)

Rajpurohit, A. S., Reylé, C., Allard, F., et al. 2014, A&A, 564, A90 (DOI: 10.1051/0004-6361/201322881)

Rajpurohit, A. S., Allard, F., Rajpurohit, S., et al. 2018, A&A, 620, A180 (DOI: 10.1051/0004-6361/201833500)

Recio-Blanco, A., de Laverny, P., Kordopatis, G., et al. 2014, A&A, 567, A5 (DOI: 10.1051/0004-6361/201322944)

Reid, I. N., & Gizis, J. E. 1997, AJ, 114, 1992 (DOI: 10.1086/118620)

Reipurth B. & Mikkola S. 2012, Nature, 492, 221  (DOI: 10.1038/nature11662 )

Rocha-Pinto H.J., &  Maciel, W.J. 1998, A&A, 339, 79

Schönrich, R., Binney, J. & Dehnen, W. 2010, MNRAS, 403, 1829 (DOI: 10.1111/j.1365-2966.2010.16253.x)

Simpson, J. D., Martell, S. L., Da Costa, G., et al. 2019, MNRAS, 482, 5302  (DOI: 10.1093/mnras/sty3042)

Tokovinin A. 2017, MNRAS, 468, 3461  (DOI: 10.1093/mnras/stx707)

Valenti, J. A., Piskunov, N. E., & Johns-Krull, C. M. 1998, ApJ, 498, 851 (DOI: 10.1086/305587)

Veyette, M. J., Muirhead, P. S., Mann, A. W., et al. 2017, ApJ, 851, 26 (DOI: 10.3847/1538-4357/aa96aa)

West, A. A., Morgan, D. P., Bochanski, J. J., et al. 2011, AJ, 141, 97 (DOI: 10.1088/0004-6256/141/3/97)

Wood, M. L.,  Mann, A. W., & Kraus, A. L.  2021, AJ, 162,128 (DOI: 10.3847/1538-3881/ac0ae9)

Woolf, V. M. &  Wallerstein, G. 2020, MNRAS, 494, 2718 (DOI: 10.1093/mnras/staa878)

Yan, Y.,  Du, C.,  Liu, S. et al. 2019, ApJ,  880, 36 (DOI:  10.3847/1538-4357/ab287d)

Zhang, S., Luo, A-Li., Comte, G., et al. 2021, ApJ, 908, 131 (DOI: 10.3847/1538-4357/abcfc5)

Ziegler, C., Tokovinin, A., Briceno, C., et al. 2020, AJ, 159, 19 (DOI: 10.3847/1538-3881/ab55e9)


# Adaptive Split Learning over Energy-Constrained Wireless Edge Networks

Zuguang Li12, Wen Wu1${}^{\textrm{,\;\Letter}}$, Shaohua Wu23, and Wei Wang4

Email: {lizg01<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Frontier Research Center, Peng Cheng Laboratory, China  2School of
Electronics and Information Engineering, Harbin Institute of Technology,
Shenzhen, China  3Department of Broadband Communication, Peng Cheng
Laboratory, China  4College of Electronic and Information Engineering, Nanjing
University of Aeronautics and Astronautics, China

###### Abstract

Split learning (SL) is a promising approach for training artificial
intelligence (AI) models, in which devices collaborate with a server to train
an AI model in a distributed manner, based on a same fixed split point.
However, due to the device heterogeneity and variation of channel conditions,
this way is not optimal in training delay and energy consumption. In this
paper, we design an adaptive split learning (ASL) scheme which can dynamically
select split points for devices and allocate computing resource for the server
in wireless edge networks. We formulate an optimization problem to minimize
the average training latency subject to long-term energy consumption
constraint. The difficulties in solving this problem are the lack of future
information and mixed integer programming (MIP). To solve it, we propose an
online algorithm leveraging the Lyapunov theory, named OPEN, which decomposes
it into a new MIP problem only with the current information. Then, a two-layer
optimization method is proposed to solve the MIP problem. Extensive simulation
results demonstrate that the ASL scheme can reduce the average training delay
and energy consumption by 53.7% and 22.1%, respectively, as compared to the
existing SL schemes.

${\textrm{\Letter}}$${\textrm{\Letter}}$footnotetext: Wen Wu<EMAIL_ADDRESS>is the corresponding author of this paper.

##  I Introduction

The coming sixth generation (6G) network is expected to evolve from connecting
people and things to connecting intelligence [1]. At the same time, as the
burgeoning edge devices and wireless sensing technology generate big data,
edge computing will become an engine for edge network intelligence [2]. The
edge network intelligence deeply integrating wireless communications and
artificial intelligence (AI) enables the edge devices for training AI models
[3, 4]. Privacy concerns and limited communication resources hinder edge
devices from offloading their data to a central server for training AI models
[5]. Split learning (SL) is a promising approach to edge network intelligence,
in which each participant trains only on its segment using its local data and
transmits only the small-scale intermediate results at the split point rather
than raw data. Therefore, SL can mitigate the aforementioned privacy and
communication concerns to a significant extent [6].

In recent years, SL has triggered a fast-growing research interest. In [7],
the per-layer execution time and energy consumption in different deep neural
networks (DNNs) were analyzed, and then the optimal point could be selected
for the best latency or best mobile energy. In [8], an architecture called
splitfed learning was proposed to address the issues in federated learning
(FL) and SL, where all clients train their local device-side model in parallel
and the fed server conducts the aggregation of the client-side local models.
In [9] a multi-split machine learning framework was proposed which reimagines
the multi-split problem as a min-cost graph search task and optimally selects
split points across multiple computing nodes to minimize the cumulative system
latency. Similarly, in [10], a split FL framework was designed to enhance
communication efficiency for splitfed learning, where both the selection of
split points and bandwidth allocation are jointly optimized to minimize the
overall system latency.

However, in practical systems, due to the device heterogeneity and variation
of channel conditions, static split points are adopted for the devices
participating in model training, which is not optimal. In addition, the energy
consumption performance should be considered in a system with constrained
energy. Thus, in wireless edge networks, it is necessary to determine optimal
split points for devices and computing resource allocation for the server, to
minimize the training delay while satisfying long-term energy consumption
constraint. The difficulties of this problem are multiple-fold. Firstly, split
point selection should dynamically adapt to the heterogeneous computing
capabilities and uncertain channels during each training iteration. Secondly,
guaranteeing long-term energy consumption constraint requires future network
information, e.g., channel conditions of devices and energy overhead of the
system. Such information is unknown beforehand.

In this paper, firstly, we design an adaptive split learning (ASL) scheme
aiming to address the device heterogeneity and channel uncertainty, and thus
to reduce the model training delay. In ASL, the heterogeneous devices
collaborate with an edge server to train an AI model. Secondly, we formulate a
joint optimization problem of the split point selection and computing resource
allocation to minimize the average system latency subject to a long-term
energy consumption constraint. Thirdly, considering that future network
information is required to solve this problem, we propose an Online sPlit
point selection and computing rEsource allocatioN (OPEN) algorithm, which
jointly determines the split point selection and computing resource allocation
in an online manner only with current information. The OPEN algorithm based on
the Lyapunov theory transforms the problem into a new problem only with the
current information, and a two-layer optimization method is proposed to solve
it. Finally, extensive simulations evaluate the performance of the proposed
scheme. The main contributions of this work are summarized as follows:

  * •

We propose an ASL scheme to reduce the model training delay in energy-
constrained wireless edge networks, which addresses the device heterogeneity
and channel uncertainty.

  * •

We design an online algorithm (i.e., OPEN algorithm) to jointly determine the
optimal split point and computing resource allocation decisions, and keep the
average energy consumption within a threshold.

The remainder of this paper is organized as follows. Section II introduces the
system model and formulates the problem. We elaborate on the OPEN algorithm in
Section III. Section IV presents the simulation results. Finally, Section V
concludes this paper.

##  II System Model and Problem Formulation

###  II-A Considered Scenario

In this paper, a typical SL scenario is considered, in which multiple mobile
devices (MDs) collaborate with a base station (BS) to train an AI model. Each
participant only trains a portion of the whole AI model, as depicted in Fig.
1. To complete model training process, the execution _procedure_ between the
BS and MDs is as follows.

Fig. 1: System model.

First, the BS transmits the device-side model to a chosen MD, which then
trains its own device-side model using its local data until reaching the split
point, thereby producing smashed data. Second, this MD transmits the smashed
data over the wireless channel to the BS, where it serves as input for the
forward propagation on the remaining model layers. Third, at the BS, the edge
server computes the loss by comparing the actual labels with the model’s
predictions and subsequently conducts backward propagation from the last layer
up to the split point, yielding the gradient of the smashed data. Fourth, the
BS relays the gradient of the smashed data at the split point back to the MD,
enabling the MD to perform the backward propagation for its device-side model
based on this gradient. Finally, upon completion of the local update, the MD
transmits the updated device-side model to the BS, whereupon the device-side
and server-side models are merged into an updated global model. The steps
above are repeated for all the participating MDs. A training episode refers to
all the devices that have been trained one time. Multiple training episodes
are performed until a satisfactory model performance is achieved.

The MDs are trained in a line manner, which are indexed by set
$\mathcal{M}=\\{1,2,...,M\\}$. The model training episode is indexed by
$n\in\mathcal{N}=\\{1,2,...,N\\}$. For the $m$-th MD participating in the SL,
the split point decision at episode $n$ is denoted by
$s_{m,n}\in\mathcal{S}=\\{1,2,...,S\\},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}$, where $\mathcal{S}$ represents the
available split points in the considered AI model. This means that the front
$s_{m,n}$ layers of the AI model are trained on the $m$-th MD, while the
remained layers are trained on the edge server. To guarantee the computing
resource constraint, the edge computing resource allocated to the $m$-th MD at
episode $n$ is denoted by $c_{m,n}$, satisfying $0\leq c_{m,n}\leq
1,~{}\forall m\in\mathcal{M},~{}n\in\mathcal{N}$.

###  II-B Training Delay

Executing SL requires model computation at the MD and BS and data exchange
between them, which incurs computation delay and transmission delay. Both
delay components are analyzed as follows.

1) Computation delay: In each local update, the device-side model and the
server-side model are trained at the MD and BS, respectively, whose delays are
analyzed as follows:

Device-side model computation delay: Let $\eta_{D}\left(s_{m,n}\right)$ denote
the total computation workload of the $m$-th MD’s device-side model at
training episode $n$, where the amount of computation workload can be
represented as the number of floating point operations (FLOPs) [11, 10].
Therefore, the computation delay of the device-side model can be defined as:

$d^{D,C}_{m,n}=\frac{\eta_{D}\left(s_{m,n}\right)}{F_{m}^{D}\delta_{m}^{D}\sigma_{m}^{D}},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.$ (1)

Here, $F_{m}^{D}$ represents the processing unit frequency of the $m$-th MD,
$\delta_{m}^{D}$ is the number of FLOPs processed by the $m$-th MD in a
processing unit cycle depending on the architecture of the processor [12], and
$\sigma_{m}^{D}$ is the number of cores the $m$-th MD has.

Server-side model computation delay: Let $\eta$ denote the total computation
workload of the model. Consequently, the server-side model’s computation
workload is the difference, i.e., $\eta-\eta_{D}\left(s_{m,n}\right)$. Similar
to that in (1), the computation delay of the server-side model is given by

$d_{m,n}^{B,C}=\frac{\eta-\eta_{D}\left(s_{m,n}\right)}{c_{m,n}F^{B}\delta^{B}\sigma^{B}},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N},$ (2)

where $F^{B}$, $\delta^{B}$, and $\sigma^{B}$ represent the frequency of the
computing server at the BS, the number of FLOPs processed by a cycle, and the
number of cores, respectively. Here, $c_{m,n}F^{B}$ is the amount of allocated
edge computing resources for the $m$-th MD.

2) Transmission delay: In SL, necessary data, MDs and the BS exchange
essential data (e.g., device-side models, smashed data, and gradients) which
incur transmission delays. The following provides a detailed analysis.

Device-side model downloading delay: To cooperatively train the AI model with
each MD, BS needs to send the latest device-side model to the MDs. Let
$R^{B}_{m,n}=W^{B}\log_{2}(1+({P^{B}\left|g^{B}_{m,n}\right|^{2}})/({(N_{0}+I)W^{B}}))$
represent the transmission rate from the BS to the $m$-th MD at episode $n$.
Here, $W^{B}$, $P^{B}$, and $g^{B}_{m,n}$ represent the downlink bandwidth,
the transmission power at the BS, and the channel coefficient between BS and
the $m$-th participating MD at episode $n$, respectively. $N_{o}$ and $I$
denote the power spectrum density of thermal noise and interference caused by
other BSs, respectively. Denote the data size of the device-side model with
split point $s_{m,n}$ as $\xi\left(s_{m,n}\right)$, which can be quantified by
the count of parameters within that portion of the model. Hence, we have the
device-side model downloading delay as follows:

$d_{m,n}^{B,D}=\frac{\xi\left(s_{m,n}\right)}{R^{B}_{m,n}},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.$ (3)

Smashed data transmission delay: In each local update, MDs send the smashed
data to the BS for the forward propagation of the server-side model. The
transmission rate from the $m$-th participating MD to the BS is represented by
$R^{D}_{m,n}=W^{D}_{m}\log_{2}(1+({P^{D}_{m}\left|g^{D}_{m,n}\right|^{2}})/({(N_{0}+N)W^{D}_{m}}))$,
where $W^{D}_{m}$, $g^{D}_{m,n}$, and $P^{D}_{m}$ represent the bandwidth, the
channel coefficient from the $m$-th MD to the BS at episode $n$, and MD’s
transmission power in the uplink transmission, respectively. In addition, the
size of the smashed data at the split point $s_{m,n}$ is $\beta(s_{m,n})$,
which is measured by the number of parameters. Thus, the smashed data
transmission delay is given by

$d_{m,n}^{D,S}=\frac{\beta(s_{m,n})}{R^{D}_{m,n}},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.$ (4)

Gradient transmission delay: The BS needs to send the smashed data’s gradient
to MDs for the device-side model’s backward propagation. Let $\gamma(s_{m,n})$
denote the data size of smashed data’s gradient given the split point
$s_{m,n}$, which is also measured by the number of parameters. The
corresponding delay is calculated as follows:

$d_{m,n}^{B,G}=\frac{\gamma(s_{m,n})}{R^{B}_{m,n}},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.$ (5)

Device-side model uploading delay: After a few local updates, the MD needs to
upload the latest device-side model to the BS for training the AI model with
the next MD. The corresponding delay is given by

$d_{m,n}^{D,D}=\frac{\xi\left(s_{m,n}\right)}{R^{D}_{m,n}},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.$ (6)

3) Overall model training delay: Taking all the computation and communication
delay components into account, the model training delay taken for the $m$-th
participating MD at each episode is given by

$\begin{split}D\left(s_{m,n},c_{m,n}\right)&=d^{D,C}_{m,n}+d^{B,C}_{m,n}+d^{B,D}_{m,n}+d^{D,S}_{m,n}\\\
&+d^{B,G}_{m,n}+d^{D,D}_{m,n},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.\end{split}$ (7)

The performance of a model is intricately linked to the batch size and number
of episodes used during training. A model requires approximately 100 episodes
to achieve satisfactory performance, especially when there are a large amount
of participating MDs. The overall model training times, denoted by $MN$, can
be considered as long terms. The average model training delay across all MDs
and the BS until a satisfactory model performance is achieved, defined as
$\overline{D}$, can be represented by

$\overline{D}=\lim_{MN\to\infty}\frac{1}{MN}\sum_{n=1}^{N}\sum_{m=1}^{M}D\left(s_{m,n},c_{m,n}\right).$
(8)

###  II-C Energy Consumption

In SL, all participating MDs need to work with the edge server to complete the
model training, which imposes a heavy energy consumption burden on MDs, the
BS, and its edge server. The split point selections and computation resource
allocation decisions will incur different energy consumption in this system.
The energy consumption of the system consists of the components as follows.

1) Transmission-related energy consumption: It is caused by data transmission,
including device-side model, smashed data, and gradients. The energy
consumption of the $m$-th MD is a product to the MD’s transmission power and
communication period, i.e.,

$E^{D,T}_{m,n}=P_{m}^{D}\left(d_{m,n}^{D,S}+d_{m,n}^{D,D}\right).$ (9)

Similarly, the energy consumption of the BS is denoted as

$E^{B,T}_{m,n}=P^{B}\left(d_{m,n}^{B,D}+d_{m,n}^{B,G}\right).$ (10)

2) Computation-related energy consumption: It is caused by the model training.
The energy consumption for each central processing unit (CPU) cycle can be
represented by $\kappa F^{2}$, where $\kappa$ reflects the effective switched
capacitance, a parameter that depends on the chip architecture [13, 14]. The
energy consumption for device-side and sever-side model training can be
defined as

$\begin{split}E^{D,C}_{m,n}&=\kappa\delta_{m}^{D}\sigma_{m}^{D}\left(F_{m}^{D}\right)^{2}\eta_{D}\left(s_{m,n}\right),\end{split}$
(11)

and

$\begin{split}E^{B,C}_{m,n}&=\kappa
c_{m,n}\delta^{B}\sigma^{B}\left(F^{B}\right)^{2}\left(\eta-\eta_{D}\left(s_{m,n}\right)\right).\end{split}$
(12)

3) Overall model energy consumption: All energy consumption of the $m$-th MD
at episode $n$ is $E^{D}_{m,n}$, denoted as
$E^{D}_{m,n}=E^{D,T}_{m,n}+E^{D,C}_{m,n}$. For the BS, the overall energy
consumption at episode $n$ is $E^{B}_{m,n}$, denoted as
$E^{B}_{m,n}=E^{B,T}_{m,n}+E^{B,C}_{m,n}$. Taking all energy consumption of
the $m$-th MD and BS into account, the overall energy consumption at each
episode is given by

$\begin{split}E(s_{m,n},c_{m,n})&=E^{D}_{m,n}+E^{B}_{m,n},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.\end{split}$ (13)

Similar to (8), the average energy consumption of the MDs and BS until
achieving a satisfactory model performance, defined as $\overline{D}$, can be
represented by

$\overline{E}=\lim_{MN\to\infty}\frac{1}{MN}\sum_{n=1}^{N}\sum_{m=1}^{M}E(s_{m,n},c_{m,n}).$
(14)

In the following optimization problem, we keep the average energy consumption
within a threshold to save the edge energy.

###  II-D Problem Formulation

We jointly determine split point selection and edge computing resource
allocating decisions to minimize the average model training delay while
satisfying the average energy consumption requirement, which is formulated as
the following problem:

$\begin{split}{\mathbf{P}_{1}:\;}\underset{\\{s_{m,n},c_{m,n}\\}}{\text{min}}~{}&\overline{D}\\\
\text{s.t.}~{}&\overline{E}\leq E_{th},\\\ &~{}0\leq c_{m,n}\leq 1,~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N},\\\ &~{}s_{m,n}\in\mathcal{S},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.\end{split}$ (15)

Here, the first constraint requires that the system’s average energy
consumption does not surpass upper limit $E_{th}$. The second and third
constraints guarantee the feasibility of edge computing resource allocation
and split point selection.

Problem $\mathbf{P}_{1}$ is a long-term stochastic optimization problem due to
the time-varying channel conditions. In addition, obtaining the optimal policy
requires the complete future information, which is impossible. Even if the
future information is available, the problem is a mixed integer programming
(MIP) since the split point selection is discrete whereas the edge computing
resource allocation is continuous.

##  III OPEN Algorithm

###  III-A Problem Transformation

The primary obstacle in tackling problem $\mathbf{P}_{1}$ lies in handling the
long-term constraints. To overcome this hurdle, we employ the Lyapunov
optimization method [15, 16]. The essence of this method involves constructing
energy deficit queues to represent the fulfillment status of long-term energy
consumption constraints, thereby steering the system towards adhering to these
constraints. The process of problem transformation is presented as follows.

First, let $T$ denote the total number of model training times between the BS
and MDs, i.e., $T=MN$, indexed by $t\in\mathcal{T}=\\{1,2,3,...,T\\}$. To
satisfy the energy consumption limitation, we introduce an energy deficit
queue for the system with its dynamic evolves as follows:

$Q^{t+1}_{m,n}=\left[Q^{t}_{m,n}+E^{t}_{m,n}-E_{th}\right]^{+},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}.$ (16)

Here, $Q^{t}_{m,n}$ indicates the queue backlog in time slot $t$ representing
the accumulated part of current energy consumption that exceeds the upper
limit, and $E^{t}_{m,n}$ is the corresponding overall energy consumption
$E(s_{m,n},c_{m,n})$ in the time slot $t$. The initial state is set to
$Q^{0}_{m,n}=0$ and the queue is updated according to (16).

######  Theorem 1.

Equation (16) is essentially equivalent to the long-term throughput constraint
in (15), if the stability condition
$\lim_{T\to\infty}Q^{T}_{m,n}/T=0,~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N}$ can be satisfied.

_Proof._ Please refer to the Proof of Theorem 1 in an online appendix [17].
$\blacksquare$

Second, we utilize a Lyapunov function, defined as
$L(Q^{t}_{m,n})={(Q^{t}_{m,n})^{2}}/2$, to capture the fulfillment status of
the long-term energy consumption constraint [18]. The Lyapunov function serves
as a quantitative indicator of congestion across all queues. A smaller value
of $L(Q^{t}_{m,n})$ signifies a lesser queue backlog, thus indicating enhanced
stability of the virtual queue. $L(Q^{t}_{m,n})$ is large when queue backlogs
are elevated, resulting in more flows experiencing throughput below the
relevant required values. To ensure the energy deficit queue remains stable,
i.e., to continuously enforce the energy consumption constraints by driving
down the Lyapunov function, we introduce one-shot Lyapunov drift, defined as

$\bigtriangleup(Q^{t}_{m,n})=\mathbb{E}[L(Q^{t+1}_{m,n})-L(Q^{t}_{m,n})|Q^{t}_{m,n}].$
(17)

Then, we have

$\begin{split}\bigtriangleup(Q^{t}_{m,n})&=\frac{1}{2}\mathbb{E}\left[(Q^{t+1}_{m,n})^{2}-(Q^{t}_{m,n})^{2}|Q^{t}_{m,n}\right]\\\
&\leq\frac{1}{2}\mathbb{E}\left[\left(Q^{t}_{m,n}+E_{m,n}^{t}-E_{th}\right)^{2}-(Q^{t}_{m,n})^{2}|Q^{t}_{m,n}\right]\\\
&=B_{1}+Q^{t}_{m,n}\mathbb{E}\left[\left(E_{m,n}^{t}-E_{th}\right)|Q^{t}_{m,n}\right]\\\
&\leq
B_{2}+Q^{t}_{m,n}\mathbb{E}\left[\left(E_{m,n}^{t}-E_{th}\right)|Q^{t}_{m,n}\right],\end{split}$
(18)

where $B_{1}=\left(E_{m,n}^{t}-E_{th}\right)^{2}/2$, and
$B_{2}=\left(E_{m,n}^{\max}-E_{th}\right)^{2}/2$ is a constant.
$E_{m,n}^{\max}$ is the maximum energy that can be produced by the $m$-th MD
and BS at episode $n$. The first inequality in (18), is due to
$(Q^{t}_{m,n})^{2}\leq\left(Q^{t}_{m,n}+E_{m,n}^{t}-E_{th}\right)^{2}$, and
the second inequality is because $E_{m,n}^{t}\leq E_{m,n}^{\max}$.

Third, leveraging the Lyapunov optimization theory, the original problem
$\mathbf{P}_{1}$ can be reformulated into minimizing a drift-plus-cost at each
time slot, as expressed below:

$\begin{split}&\bigtriangleup(Q^{t}_{m,n})+V\mathbb{E}\left[D^{t}_{m,n}|Q^{t}_{m,n}\right]\\\
&\leq
B_{2}+Q^{t}_{m,n}\mathbb{E}\left[\left(E_{m,n}^{t}-E_{th}\right)|Q^{t}_{m,n}\right]+V\mathbb{E}\left[D^{t}_{m,n}|Q^{t}_{m,n}\right],\end{split}$
(19)

Here, $D^{t}_{m,n}$ represents the overall energy consumption
$D(s_{m,n},c_{m,n})$ at the time slot $t$, and $V$ is a positive and
adjustable parameter that adjusts the balance between the training delay and
energy consumption. The inequality in (19) is due to the upper limit in (18).

Because both $B_{2}$ and $E_{th}$ are constants, the original problem
$\mathbf{P}_{1}$ can be reformulated into a new optimization problem
$\mathbf{P}_{2}$:

$\begin{split}{\mathbf{P}_{2}:\;}\underset{\\{s_{m,n}^{t},c_{m,n}^{t}\\}_{\begin{subarray}{c}\forall
m\in\mathcal{M},\\\ ~{}n\in\mathcal{N}\end{subarray}}}{\text{min}}~{}~{}&f\\\
\text{s.t.}~{}&0\leq c_{m,n}^{t}\leq 1,~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N},\\\ &s_{m,n}^{t}\in\mathcal{S},~{}\forall
m\in\mathcal{M},~{}n\in\mathcal{N},\end{split}$ (20)

where $f=VD_{m,n}^{t}+Q^{t}_{m,n}E_{m,n}^{t}$. It is worth noting that solving
problem $\mathbf{P}_{2}$ only requires the current information as input.

Problem $\mathbf{P}_{2}$ is a non-convex optimization problem, in which two
decision variables are mixed integers and the objective function is non-
convex. To solve the MIP problem, we propose a two-layer optimization method
to jointly determine the optimal split point and computing resource allocation
decisions, which is detailed as follows.

###  III-B Upper Layer Analysis

When the split point selecting decision is fixed, the edge computing resource
allocation problem can be transformed into an upper-layer problem as follows:

$\begin{split}{\mathbf{P}_{3}:\;}{\text{min}}~{}~{}&f\\\ \text{s.t.}~{}&0\leq
c_{m,n}^{t}\leq 1.\end{split}$ (21)

######  Theorem 2.

For a given split point selecting decision $\\{s_{m,n}^{t}\\}$, the optimal
solution for problem $\mathbf{P}_{3}$ is given by

$(c_{m,n}^{t})^{\star}=\begin{cases}1,\hfill\text{if
}\sqrt{\frac{V\omega_{1}}{\omega_{4}Q^{t}_{m,n}}}>1,\\\
\sqrt{\frac{V\omega_{1}}{\omega_{4}Q^{t}_{m,n}}},\hfill\text{otherwise},\end{cases}$
(22)

where
$\omega_{1}={\left(\eta-\eta_{D}\left(s_{m,i}^{t}\right)\right)}/{\left(F^{B}\delta^{B}\sigma^{B}\right)}$,
and
$\omega_{4}=\kappa\delta^{B}\sigma^{B}\left(F^{B}\right)^{2}\left(\eta-\eta_{D}\left(s_{m,i}^{t}\right)\right)$.

_Proof._ Please refer to the Proof of Theorem 2 in the online appendix.
$\blacksquare$

###  III-C Lower layer Analysis

When the optimal computing resource allocation is given, the split point
selection problem can be transformed into a lower-layer problem as follows:

$\begin{split}{\mathbf{P}_{4}:\;}{\text{min}}~{}~{}&f\\\
\text{s.t.}~{}&s_{m,n}^{t}\in\mathcal{S}.\end{split}$ (23)

According to (7) and (13), the objective function in (23) can be rewritten as

$\begin{split}f\left(s_{m,n}^{t}\right)&=VD_{m,n}^{t}+Q^{t}_{m,n}E_{m,n}^{t}\\\
&=V\left(d^{D,C}_{m,n}+d^{B,C}_{m,n}+d^{B,D}_{m,n}+d^{D,S}_{m,n}+d^{B,G}_{m,n}+d^{D,D}_{m,n}\right)\\\
&~{}~{}+Q^{t}_{m,n}\left(E^{B,T}(s_{m,n}^{t})+E^{B,C}(s_{m,n}^{t},c_{m,n}^{t})\right).\end{split}$
(24)

The objective function in problem $\mathbf{P}_{4}$ is non-convex because the
data sizes (e.g., device-side model, smashed data, and its gradient) are
arbitrary functions in terms of the split point. This renders an analytical
closed-form expression for the optimal split point infeasible, necessitating
numerical methods for its calculation. Given the finite number of potential
split points within the AI model, the lower-layer problem can be tackled using
an exhaustive searching method.

The OPEN algorithm is shown in Algorithm 1. In the algorithm, the local
optimal decision of the edge computing resource allocation is determined based
on (22), and the local optimal split point is obtained by searching those
split points to minimize $f\left(s_{m,n}^{t}\right)$. Then, the optimal split
point and computing resource allocation decisions can be determined through
several iterations.

Algorithm 1 OPEN algorithm

1:$Q^{0}_{m,n}\leftarrow
0,F_{m}^{D},\delta_{m}^{D},\sigma_{m}^{D},F^{B},\delta^{B},\sigma^{B},P_{m}^{D},$
$P^{B},E_{th}$;

2:split point selecting decisions $(s_{m,n}^{t})^{\star}$, and computing
resource allocating decisions $(c_{m,n}^{t})^{\star}$;

3:Initiate $R_{m,n}^{D,t}$, $R_{m,n}^{B,t}$, $f_{min}$, $Q^{t-1}_{m,n}$;

4:Set $(c_{m,n}^{t})^{\star}=1$, and $(s_{m,n}^{t})^{\star}=0$;

5:while $||(c_{m,n}^{t})^{\star}-c_{m,n}^{t,last}||>0.01$ and
$(s_{m,n}^{t})^{\star}\neq s_{m,n}^{t,last}$ do

6:
$c_{m,n}^{t,last}=(c_{m,n}^{t})^{\star},s_{m,n}^{t,last}=(s_{m,n}^{t})^{\star}$;

7: Compute $(c_{m,n}^{t})^{\star}$ based on (22);

8: for $s_{m,n}^{t}=1$ to $S$ do

9: Compute $D_{m,n}^{t}$ based on (7);

10: Compute $E_{m,n}^{t}$ based on (13);

11: $Q^{t}_{m,n}=\max\\{Q^{t-1}_{m,n}+E^{t}_{m,n}-E_{th},0\\}$;

12: Compute $f\left(s_{m,n}^{t}\right)$ based on (24);

13: if $f\left(s_{m,n}^{t}\right)<f_{min}$ then

14: $f_{min}=f\left(s_{m,n}^{t}\right)$;

15: $(s_{m,n}^{t})^{\star}=s_{m,n}^{t}$;

16: end if

17: end for

18:end while

19:return $(s_{m,n}^{t})^{\star},(c_{m,n}^{t})^{\star}$

##  IV Simulation Results

###  IV-A Simulation Setup

The time-varying wireless channel gain of the $m$-th MD, i.e., $g^{D}_{m,n}$,
is derived from a Rayleigh fading model as
$g^{D}_{m,n}=\rho^{D}_{m,n}\bar{g}^{D}_{m}$ [19]. Here, $\rho^{D}_{m,n}$ is an
independent random channel fading factor following an exponential distribution
with unity mean. Meanwhile, $\bar{g}^{D}_{m}$ is the average channel gain
calculated by using the free-space path loss model as
$\bar{g}^{D}_{m}=A_{m}^{D}\left(\frac{3\cdot 10^{8}}{4\pi
f_{m}^{D}d_{m}}\right)^{\phi_{m}^{D}}$. Here, $A_{m}^{D}$, $f_{m}^{D}$, and
$\phi_{m}^{D}$ are the antenna gain, the carrier frequency, and the path loss
exponent of the $m$-th MD, respectively, while $d_{m}$ is the distance between
the $m$-th MD and BS. Analogous to $g^{D}_{m,n}$, the BS’s time-varying
wireless channel gain $g^{B}_{m,n}$ also adheres to the same fading channel
model, but it incorporates different values for some parameters.

In the simulation, we adopt a 12-layer LeNet model [20] with the widely-
adopted MNIST data. The batch size of the training model is set to be $16$.
Due to the small amount of computation workload and parameters in the
activation layers (ReLU), we restrict the selection of split points to only
include convolution layers and fully-connected layers. Thus, there are 12
available split points in the LeNet model.

We consider 30 MDs and one BS with an edge server, as illustrated in Fig. 1.
Here, the MDs’ processing unit frequencies follow a uniform distribution
within $\left[0.5,3\right]$ GHz, the numbers of cores the MDs have follow a
uniform distribution within $\left[1,8\right]$, and the distances between MDs
and the BS follow a uniform distribution within $\left[100,1000\right]$
meters. Specially, the processing unit frequencies of MD 1, MD 2, MD 3, and MD
4 are set to 0.5 GHz, 1 GHz, 2 GHz, and 4 GHz, their numbers of cores are 1,
4, 8, and 16, respectively, and their distances are all set to 200 meters. The
other main simulation parameters are concluded in Table I.

TABLE I: Simulation parameters. Parameter | Value | Parameter | Value
---|---|---|---
$A_{m}^{D}$ | 4.11 | $A_{m}^{B}$ | 8
$\delta_{m}^{D}$ | 8 | $\delta^{B}$ | 16
$\sigma^{B}$ | 32 | $\kappa$ | $10^{-26}$
$\phi_{m}^{D}$ | 1 | $\phi_{m}^{B}$ | 1
$F^{B}$ | 3 GHz | $E_{th}$ | 3,000 J
$f_{m}^{D}$ | 2,000 MHz | $f_{m}^{B}$ | 2,000 MHz
$P_{m}^{D}$ | 0.4 W | $P^{B}$ | 3 W
$W_{m}^{D}$ | 20 MHz | $W^{B}$ | 40 MHz
$N_{o}$ |  $-174$ dBm/Hz | $N$ |  $-164$ dBm/Hz

###  IV-B Performance Evaluation

We evaluate the optimal split points and computing resource allocation
decisions on the MDs with different computing power. As shown in Fig. 2a, we
can observe that the optimal split points of MDs dynamically change with
episodes, but each of them maintains a split point in general. Furthermore,
with the progressive enhancement of an MD’s computing capability, its optimal
split point shifts closer to the output end. Consequently, it is a lesser need
for allocating computing resources, as depicted in Fig. 2b. This is because
the MD with strong computing power prefers training the majority of the AI
model locally rather than delegating it to the edge server, which benefits
reducing the transmission delay.

(a) Optimal split point.

(b) Optimal resource allocation decision.

Figure 2:  Optimal split point and computing resource allocation at each
episode.

We compare the ASL scheme with the following benchmark schemes: (1) SL, where
all devices train the AI model before split point 9 and the edge server
releases the total computing resource; (2) Delay-optimal, where the goal is to
minimize the overall model training delay; (3) Energy-optimal, where
minimizing the overall energy consumption is the goal. As shown in Fig. 3a,
the delays under different schemes fluctuate with the model training, but all
remain stable in general. We can observe that the proposed ASL scheme achieves
53.7% and 62.5% reduction in the average training delay, as compared to the SL
and energy-optimal schemes, respectively. Although the training delay of
delay-optimal scheme is smaller than that of the ASL scheme, its energy
consumption is much larger than that of the ASL scheme, as shown in Fig. 3b.
In addition, from Fig. 3b, we can see that the energy consumption of the ASL
scheme is always maintained below the upper limit, i.e., 3,000 J. Compared to
SL scheme, the average energy consumption of the proposed ASL scheme is
reduced by 22.1%.

(a) Delay across all episodes.

(b) Energy consumption across all episodes.

Figure 3: Delay and energy consumption performance with respect to different
schemes.

##  V Conclusion

In this paper, we have designed a novel ASL scheme for deploying the AI model
in energy-constrained wireless edge networks. We have formulated a problem to
minimize long-term system latency subject to a long-term energy consumption
constraint. To solve this problem, we have proposed an online algorithm to
make the optimal split point and computing resource allocation decisions.
Extensive simulation results have demonstrated the effectiveness of the ASL
scheme in reducing training latency. Due to low training latency and
acceptable energy consumption, the ASL scheme can be applied to facilitate AI
model training in energy-constrained wireless networks. For future work, we
will investigate the split point selection problem in non-linear AI models.

## References

  * [1] X. Shen, J. Gao, W. Wu, M. Li, C. Zhou, and W. Zhuang, “Holistic network virtualization and pervasive network intelligence for 6G,” _IEEE Commun. Surveys Tuts._ , vol. 24, no. 1, pp. 1–30, 2022.
  * [2] S. Deng, H. Zhao, W. Fang, J. Yin, S. Dustdar, and A. Y. Zomaya, “Edge intelligence: The confluence of edge computing and artificial intelligence,” _IEEE Internet Things J._ , vol. 7, no. 8, pp. 7457–7469, 2020.
  * [3] W. Xu, Z. Yang, D. W. K. Ng, M. Levorato, Y. C. Eldar, and M. Debbah, “Edge learning for B5G networks with distributed signal processing: Semantic communication, edge computing, and wireless sensing,” _IEEE J. Sel. Top. Signal Process._ , vol. 17, no. 1, pp. 9–39, 2023.
  * [4] X. Shen, J. Gao, W. Wu, K. Lyu, M. Li, W. Zhuang, X. Li, and J. Rao, “AI-assisted network-slicing based next-generation wireless networks,” _IEEE Open J. Veh. Technol._ , vol. 1, pp. 45–66, 2020.
  * [5] B. Xie, Y. Sun, S. Zhou, Z. Niu, Y. Xu, J. Chen, and D. Gunduz, “MOB-FL: Mobility-aware federated learning for intelligent connected vehicles,” in _Proc. IEEE ICC_ , 2023, pp. 3951–3957.
  * [6] W. Wu, M. Li, K. Qu, C. Zhou, X. Shen, W. Zhuang, X. Li, and W. Shi, “Split learning over wireless networks: Parallel design and resource management,” _IEEE J. Sel. Areas Commun._ , vol. 41, no. 4, pp. 1051–1066, 2023.
  * [7] Y. Kang, J. Hauswald, C. Gao, A. Rovinski, T. Mudge, J. Mars, and L. Tang, “Neurosurgeon: Collaborative intelligence between the cloud and mobile edge,” _ACM SIGARCH Computer Architecture News_ , vol. 45, no. 1, pp. 615–629, 2017.
  * [8] C. Thapa, P. C. M. Arachchige, S. Camtepe, and L. Sun, “Splitfed: When federated learning meets split learning,” in _Proc. AAAI Conf. Artif. Intell._ , vol. 36, no. 8, 2022, pp. 8485–8493.
  * [9] S. Wang, X. Zhang, H. Uchiyama, and H. Matsuda, “HiveMind: Towards cellular native machine learning model splitting,” _IEEE J. Sel. Areas Commun._ , vol. 40, no. 2, pp. 626–640, 2022.
  * [10] C. Xu, J. Li, Y. Liu, Y. Ling, and M. Wen, “Accelerating split federated learning over wireless communication networks,” _IEEE Trans. Wireless Commun._ , 2023, DOI:10.1109/TWC.2023.3327372.
  * [11] Z. Xu, F. Yu, C. Liu, and X. Chen, “Reform: Static and dynamic resource-aware DNN reconfiguration framework for mobile device,” in _Proc. ACM Des. Autom. Conf._ , 2019, pp. 1–6.
  * [12] Q. Zeng, Y. Du, K. Huang, and K. K. Leung, “Energy-efficient resource management for federated edge learning with CPU-GPU heterogeneous computing,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 12, pp. 7947–7962, 2021.
  * [13] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, “A joint learning and communications framework for federated learning over wireless networks,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 1, pp. 269–283, 2020.
  * [14] Y. Dai, D. Xu, S. Maharjan, and Y. Zhang, “Joint computation offloading and user association in multi-task mobile edge computing,” _IEEE Trans. Veh. Technol._ , vol. 67, no. 12, pp. 12 313–12 325, 2018.
  * [15] M. J. Neely, “Stochastic network optimization with application to communication and queueing systems,” _Synthesis Lectures on Communication Networks_ , vol. 3, no. 1, pp. 1–211, 2010.
  * [16] J. Luo, F. R. Yu, Q. Chen, and L. Tang, “Adaptive video streaming with edge caching and video transcoding over software-defined mobile networks: A deep reinforcement learning approach,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 3, pp. 1577–1592, 2019.
  * [17] Appendix. [Online]. Available: https://github.com/lizuguang/Appendix
  * [18] J. Xu, L. Chen, and P. Zhou, “Joint service caching and task offloading for mobile edge computing in dense networks,” in _Proc. IEEE Conf. Comput. Commun. Workshops_ , 2018, pp. 207–215.
  * [19] L. Huang, S. Bi, and Y.-J. A. Zhang, “Deep reinforcement learning for online computation offloading in wireless powered mobile-edge computing networks,” _IEEE Trans. Mob. Comput._ , vol. 19, no. 11, pp. 2581–2593, 2020.
  * [20] T. Nishio and R. Yonetani, “Client selection for federated learning with heterogeneous resources in mobile edge,” in _Proc. IEEE ICC_ , 2019, pp. 1–7.