Implementation of the multinomial generalized linear model (softmax link)

[jachymb]

Summary

This is the implementation of the multinomial_logit_glm_lpmf, including the OpenCL variant. A solution to #3149

It computes, in an efficient vectorized way, exactly what one would expect:

$\mathrm{multinomialLogitGLM} (y|x, \alpha, \beta) = \prod_n \mathrm{multinomial}(y_n \mid \mathrm{softmax}(x_n \beta + \alpha_n))$

I was trying to closely follow patterns in related functions like the binomial_logit_glm_lmpf and categorical_logit_glm_lpmf which this is a generalization of both.

The typing:

• observations y are a 2D array [,] int non-negative (rows=instances, columns=outcome classes)
• thus the predictors x are matrix(rows=instances, columns=features)
• the coefficients beta are a matrix (rows=features, columns=outcome classes)
• then the intercept alpha may be either a matrix (same shape as y) or a row_vector (length=outcome classes). Now this may seem a little irregular at first, but it makes sense when you really think about it. Other GLMs offer either a vector or real for the intercept. In the multivariate case, the (univariate) vector corresponds to the matrix and the scalar to be copied for each instance is more naturally represented here as a row_vector rather than a column. (It can be seen as an extra row of beta corresponding to a constant feature, also matching the shape of y and in consequence, the implementation algebra is just more natural this way.) However this may be argued to be a little inconsistent with categorical_logit_glm_lpmf so it's maybe up to a debate, but then the categorical is not implemented as multivariate even though it also uses beta matrix.

I chose to use "logit" in the name instead of "softmax" (even though it really is softmax) to be consistent with existing functions like multinomial_logit.

I am new to this, so if there are any improvements to be made, I will gladly listen. I tried to use efficient vectorized approach and utilize existing functions where applicable. Note however this uses its own softmax implementation, which is necessary for efficient vectorization and in my understanding can't be done by the existing library implementation the way it's needed here. Maybe if my other PR #3313 is approved, this could be made a few LOC shorter with a call to that.

As a personal note, this is a function I want available in Stan for my actual work. :)

Tests

Standard testing of the new functions introduced, both for the prim and cl variants.

Side Effects

None.

Release notes

Add multinomial_logit_glm_lpmf, including OpenCL support.

AI use disclosure

I used claude code w/ Sonnet to help with the work, but I critically reviewed/edited every single line, striving to match the general code quality and patterns within the library.

Checklist

• Copyright holder: me, jachymb@gmail.com
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested

[jachymb]

I think the failed tests are due to some server errors, may I ask for a rerun?

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_regr/gp_regr.stan	0.1	0.1	0.99	-0.63% slower
gp_regr/gen_gp_data.stan	0.03	0.03	1.0	-0.16% slower
arK/arK.stan	1.87	1.81	1.03	3.21% faster
eight_schools/eight_schools.stan	0.06	0.06	1.02	1.59% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	9.14	8.94	1.02	2.22% faster
pkpd/one_comp_mm_elim_abs.stan	20.3	20.24	1.0	0.31% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.26	0.27	0.95	-5.21% slower
sir/sir.stan	73.57	72.7	1.01	1.19% faster
gp_pois_regr/gp_pois_regr.stan	3.01	2.97	1.01	1.23% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.77	2.79	0.99	-0.66% slower
irt_2pl/irt_2pl.stan	4.43	4.42	1.0	0.42% faster
arma/arma.stan	0.31	0.31	1.0	0.25% faster
garch/garch.stan	0.46	0.45	1.02	1.84% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	0.96	-3.67% slower
performance.compilation	243.79	238.26	1.02	2.27% faster
Mean result: 1.0032624211287606

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Jenkins Console Log
Blue Ocean
Commit hash: 7a7d72e615616c0f3361edc354f888fe5bb9b0d9

---

Machine information

No LSB modules are available. Distributor ID: Ubuntu Description: Ubuntu 20.04.3 LTS Release: 20.04 Codename: focal

CPU:
Architecture: x86_64
CPU op-mode(s): 32-bit, 64-bit
Byte Order: Little Endian
Address sizes: 46 bits physical, 48 bits virtual
CPU(s): 80
On-line CPU(s) list: 0-79
Thread(s) per core: 2
Core(s) per socket: 20
Socket(s): 2
NUMA node(s): 2
Vendor ID: GenuineIntel
CPU family: 6
Model: 85
Model name: Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz
Stepping: 4
CPU MHz: 2400.000
CPU max MHz: 3700.0000
CPU min MHz: 1000.0000
BogoMIPS: 4800.00
Virtualization: VT-x
L1d cache: 1.3 MiB
L1i cache: 1.3 MiB
L2 cache: 40 MiB
L3 cache: 55 MiB
NUMA node0 CPU(s): 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78
NUMA node1 CPU(s): 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79
Vulnerability Gather data sampling: Mitigation; Microcode
Vulnerability Itlb multihit: KVM: Mitigation: Split huge pages
Vulnerability L1tf: Mitigation; PTE Inversion; VMX conditional cache flushes, SMT vulnerable
Vulnerability Mds: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Meltdown: Mitigation; PTI
Vulnerability Mmio stale data: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Reg file data sampling: Not affected
Vulnerability Retbleed: Mitigation; IBRS
Vulnerability Spec rstack overflow: Not affected
Vulnerability Spec store bypass: Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1: Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2: Mitigation; IBRS; IBPB conditional; STIBP conditional; RSB filling; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds: Not affected
Vulnerability Tsx async abort: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Vmscape: Mitigation; IBPB before exit to userspace
Flags: fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc art arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc cpuid aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 sdbg fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm 3dnowprefetch cpuid_fault epb cat_l3 cdp_l3 invpcid_single pti intel_ppin ssbd mba ibrs ibpb stibp tpr_shadow vnmi flexpriority ept vpid ept_ad fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm cqm mpx rdt_a avx512f avx512dq rdseed adx smap clflushopt clwb intel_pt avx512cd avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local dtherm ida arat pln pts hwp hwp_act_window hwp_epp hwp_pkg_req pku ospke md_clear flush_l1d arch_capabilities

G++:
g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang:
clang version 10.0.0-4ubuntu1
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin