Machine Learning Part 3 — Linear regression in SQL revisited

This is the third part of the ML series. For your convenience you can find other parts in the table of contents in Part 1 – Linear regression in MXNet

Last time we saw how to calculate linear regression for the Iris dataset. However, we had to hardcode all the featuers. Today we are going to make our query much more flexible. Let’s begin.

We start with the same schema as before. Now, the query, this time for PostgreSQL 10:

WITH RECURSIVE constants AS ( SELECT 1 AS column_1 UNION SELECT 2 AS column_1 UNION SELECT 3 AS column_1 UNION SELECT 4 AS column_1 UNION SELECT 5 AS column_1 ), extended AS ( SELECT S.*, CASE WHEN S.iris = 'setosa' THEN 1.0 ELSE 0.0 END AS is_setosa, CASE WHEN S.iris = 'virginica' THEN 1.0 ELSE 0.0 END AS is_virginica FROM samples AS S order by random() ), training AS ( SELECT * FROM extended LIMIT 100 ), test AS ( SELECT * FROM extended EXCEPT SELECT * FROM training ), numbered AS( SELECT *, ROW_NUMBER() OVER (ORDER BY (SELECT NULL)) AS row_number FROM training ), pivoted_training AS ( SELECT N.row_number AS sample, C.column_1 AS feature, CAST(CASE WHEN C.column_1 = 1 THEN N.sepal_width WHEN C.column_1 = 2 THEN N.petal_length WHEN C.column_1 = 3 THEN N.petal_width WHEN C.column_1 = 4 THEN N.is_setosa ELSE N.is_virginica END AS float) AS value, N.sepal_length AS y FROM numbered AS N, constants AS C ), learning AS ( SELECT C.column_1 AS feature, CAST(0.0 AS float) as w, CAST(0.0 AS float) as b, CAST(0.0 AS float) as gw, CAST(0.0 AS float) as gb, 1 as iteration, CAST(0.0 AS float) as mse, CAST(0.0 AS float) as distance, 1 as dummy FROM constants AS C UNION ALL SELECT R.feature, R.w, R.b, R.gw, R.gb, R.iteration, R.mse, R.distance, R.dummy FROM ( SELECT CAST(Z.w AS float) AS w, CAST(Z.b AS float) AS b, CAST(AVG(Z.gw) OVER(PARTITION BY Z.feature) AS float) AS gw, CAST(AVG(Z.gb) OVER(PARTITION BY Z.feature) AS float) AS gb, Z.iteration + 1 AS iteration, Z.feature, CAST(AVG(Z.squared_distance) OVER(PARTITION BY Z.dummy) AS float) AS mse, Z.sample AS sample, CAST(Z.distance AS FLOAT) AS distance, Z.dummy FROM ( SELECT X.*, X.distance * x.distance AS squared_distance, X.distance * X.value AS gw, X.distance AS gb FROM ( SELECT K.*, SUM(K.value * K.w + K.b) OVER(PARTITION BY K.sample) - K.y AS distance FROM ( SELECT T.*, L.w, L.b, L.iteration, L.dummy FROM pivoted_training AS T INNER JOIN ( SELECT L.w - 0.01 * L.gw AS w, L.b - 0.01 * L.gb AS b, L.feature, L.iteration, MAX(L.iteration) OVER(PARTITION BY L.dummy) AS max_iteration, L.dummy FROM learning AS L ) AS L ON T.feature = L.feature AND L.iteration = max_iteration WHERE L.iteration < 100 ) AS K ) AS X ) AS Z ) AS R WHERE R.sample = 1 ) SELECT * FROM learning 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 WITH RECURSIVE constants AS ( SELECT 1 AS column_1 UNION SELECT 2 AS column_1 UNION SELECT 3 AS column_1 UNION SELECT 4 AS column_1 UNION SELECT 5 AS column _ 1 ) , extended AS ( SELECT S . * , CASE WHEN S . iris = 'setosa' THEN 1.0 ELSE 0.0 END AS is_setosa , CASE WHEN S . iris = 'virginica' THEN 1.0 ELSE 0.0 END AS is_virginica FROM samples AS S order by random ( ) ) , training AS ( SELECT * FROM extended LIMIT 100 ) , test AS ( SELECT * FROM extended EXCEPT SELECT * FROM training ) , numbered AS ( SELECT * , ROW_NUMBER ( ) OVER ( ORDER BY ( SELECT NULL ) ) AS row_number FROM training ) , pivoted_training AS ( SELECT N . row_number AS sample , C . column_1 AS feature , CAST ( CASE WHEN C . column_1 = 1 THEN N . sepal_width WHEN C . column_1 = 2 THEN N . petal_length WHEN C . column_1 = 3 THEN N . petal_width WHEN C . column_1 = 4 THEN N . is_setosa ELSE N . is_virginica END AS float ) AS value , N . sepal_length AS y FROM numbered AS N , constants AS C ) , learning AS ( SELECT C . column_1 AS feature , CAST ( 0.0 AS float ) as w , CAST ( 0.0 AS float ) as b , CAST ( 0.0 AS float ) as gw , CAST ( 0.0 AS float ) as gb , 1 as iteration , CAST ( 0.0 AS float ) as mse , CAST ( 0.0 AS float ) as distance , 1 as dummy FROM constants AS C UNION ALL SELECT R . feature , R . w , R . b , R . gw , R . gb , R . iteration , R . mse , R . distance , R . dummy FROM ( SELECT CAST ( Z . w AS float ) AS w , CAST ( Z . b AS float ) AS b , CAST ( AVG ( Z . gw ) OVER ( PARTITION BY Z . feature ) AS float ) AS gw , CAST ( AVG ( Z . gb ) OVER ( PARTITION BY Z . feature ) AS float ) AS gb , Z . iteration + 1 AS iteration , Z . feature , CAST ( AVG ( Z . squared_distance ) OVER ( PARTITION BY Z . dummy ) AS float ) AS mse , Z . sample AS sample , CAST ( Z . distance AS FLOAT ) AS distance , Z . dummy FROM ( SELECT X . * , X . distance * x . distance AS squared_distance , X . distance * X . value AS gw , X . distance AS gb FROM ( SELECT K . * , SUM ( K . value * K . w + K . b ) OVER ( PARTITION BY K . sample ) - K . y AS distance FROM ( SELECT T . * , L . w , L . b , L . iteration , L . dummy FROM pivoted_training AS T INNER JOIN ( SELECT L . w - 0.01 * L . gw AS w , L . b - 0.01 * L . gb AS b , L . feature , L . iteration , MAX ( L . iteration ) OVER ( PARTITION BY L . dummy ) AS max_iteration , L . dummy FROM learning AS L ) AS L ON T . feature = L . feature AND L . iteration = max_iteration WHERE L . iteration < 100 ) AS K ) AS X ) AS Z ) AS R WHERE R . sample = 1 ) SELECT * FROM learning

Uuu, terrible. Let’s go step by step.

First, constants is just a table with some numbers. We have 5 features so we have 5 rows there. This could be done much easier with recursive CTE or any other dynamic solution.

Next, extended : we just add two more features and randomize the rows.

training , test , and numbered are just tables for bookkeeping the samples.

pivoted_training : here comes some magic. We don’t want to have rows with all the features inside, we want to have one row for each sample’s feature. So we do the translation and emit rows with sample number, feature id, feature value, and target variable.

Next comes our recursive CTE for training. We start with some rows representing coefficients for each feature. We initialize and , as well as iteration and distances. We have dummy column again.

Next, we do the calculation. Let’s go from the middle.

We do the training in similar way as before. Assuming we have samples, each has features, the inner join target has rows where stands for iterations. We join this with samples ( rows) based on the feature id and maximum iteration. So finally, we should have rows per each iteration.

Next, we calculate the distance for each sample (this is the PARTITION BY K.sample part).

Next, we square the distance and calculate gradient coefficients for each sample.

Finally, we cast variables and calculate the final gradients by taking averages over features. We also calculate mse and we are almost done.

The only tricky part is how to get exactly rows representing new coefficients. This is done by WHERE R.sample = 1 as for each sample we have exactly the same results so we can just take any of them.

Finally, we get our training results with SELECT * FROM learning . You can see it here:

feature w b gw gb iteration mse distance dummy 3 0 0 0 0 1 0 0 1 2 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 5 0 0 0 0 1 0 0 1 4 0 0 0 0 1 0 0 1 1 0 0 -16.52 -5 2 25.01 -5.1 1 2 0 0 -7.245 -5 2 25.01 -5.1 1 3 0 0 -0.755 -5 2 25.01 -5.1 1 4 0 0 -5 -5 2 25.01 -5.1 1 5 0 0 0 -5 2 25.01 -5.1 1 1 0.1652 0.05 -13.3746025 -4.048655 3 16.39653605105 -4.11886 1 2 0.07245 0.05 -5.8670395 -4.048655 3 16.39653605105 -4.11886 1 3 0.00755 0.05 -0.6108085 -4.048655 3 16.39653605105 -4.11886 1 4 0.05 0.05 -4.048655 -4.048655 3 16.39653605105 -4.11886 1 5 0 0.05 0 -4.048655 3 16.39653605105 -4.11886 1 1 0.298946025 0.09048655 -10.8278890377 -3.278385532 4 10.7499354232339 -3.3244694425 1 2 0.131120395 0.09048655 -4.751354825875 -3.278385532 4 10.7499354232339 -3.3244694425 1 3 0.013658085 0.09048655 -0.494062025325 -3.278385532 4 10.7499354232339 -3.3244694425 1 4 0.09048655 0.09048655 -3.278385532 -3.278385532 4 10.7499354232339 -3.3244694425 1 5 0 0.09048655 0 -3.278385532 4 10.7499354232339 -3.3244694425 1 1 0.407224915377 0.12327040532 -8.76590822437997 -2.65472632382273 5 7.04827711689888 -2.6812831026476 1 2 0.17863394325875 0.12327040532 -3.84802533060171 -2.65472632382273 5 7.04827711689888 -2.6812831026476 1 3 0.01859870525325 0.12327040532 -0.399536787514652 -2.65472632382273 5 7.04827711689888 -2.6812831026476 1 4 0.12327040532 0.12327040532 -2.65472632382273 -2.65472632382273 5 7.04827711689888 -2.6812831026476 1 5 0 0.12327040532 0 -2.65472632382273 5 7.04827711689888 -2.6812831026476 1 1 0.4948839976208 0.149817668558227 -7.09639777302516 -2.14977210051383 6 4.62163562959118 -2.16052130716148 1 2 0.217114196564767 0.149817668558227 -3.11663208541266 -2.14977210051383 6 4.62163562959118 -2.16052130716148 1 3 0.0225940731283965 0.149817668558227 -0.323003275409457 -2.14977210051383 6 4.62163562959118 -2.16052130716148 1 4 0.149817668558227 0.149817668558227 -2.14977210051383 -2.14977210051383 6 4.62163562959118 -2.16052130716148 1 5 0 0.149817668558227 0 -2.14977210051383 6 4.62163562959118 -2.16052130716148 1 1 0.565847975351051 0.171315389563366 -5.7446562719125 -1.74092897782157 7 3.03083789510468 -1.73888220332818 1 2 0.248280517418894 0.171315389563366 -2.52444935656594 -1.74092897782157 7 3.03083789510468 -1.73888220332818 1 3 0.0258241058824911 0.171315389563366 -0.261037007948566 -1.74092897782157 7 3.03083789510468 -1.73888220332818 1 4 0.171315389563366 0.171315389563366 -1.74092897782157 -1.74092897782157 7 3.03083789510468 -1.73888220332818 1 5 0 0.171315389563366 0 -1.74092897782157 7 3.03083789510468 -1.73888220332818 1 1 0.623294538070176 0.188724679341581 -4.65020071011763 -1.40990351099703 8 1.98798177898635 -1.39749913013413 1 2 0.273525010984553 0.188724679341581 -2.04498030998884 -1.40990351099703 8 1.98798177898635 -1.39749913013413 1 3 0.0284344759619768 0.188724679341581 -0.21086530760641 -1.40990351099703 8 1.98798177898635 -1.39749913013413 1 4 0.188724679341581 0.188724679341581 -1.40990351099703 -1.40990351099703 8 1.98798177898635 -1.39749913013413 1 5 0 0.188724679341581 0 -1.40990351099703 8 1.98798177898635 -1.39749913013413 1 1 0.669796545171352 0.202823714451552 -3.76406019771898 -1.14188416444708 9 1.30433157451662 -1.12109643966513 1 2 0.293974814084442 0.202823714451552 -1.65677142468737 -1.14188416444708 9 1.30433157451662 -1.12109643966513 1 3 0.0305431290380408 0.202823714451552 -0.170243238427965 -1.14188416444708 9 1.30433157451662 -1.12109643966513 1 4 0.202823714451552 0.202823714451552 -1.14188416444708 -1.14188416444708 9 1.30433157451662 -1.12109643966513 1 5 0 0.202823714451552 0 -1.14188416444708 9 1.30433157451662 -1.12109643966513 1 1 0.707437147148542 0.214242556096022 -3.04658478966489 -0.924878577539922 10 0.856160630419917 -0.897305996455662 1 2 0.310542528331315 0.214242556096022 -1.3424525664871 -0.924878577539922 10 0.856160630419917 -0.897305996455662 1 3 0.0322455614223205 0.214242556096022 -0.137353157576775 -0.924878577539922 10 0.856160630419917 -0.897305996455662 1 4 0.214242556096022 0.214242556096022 -0.924878577539922 -0.924878577539922 10 0.856160630419917 -0.897305996455662 1 5 0 0.214242556096022 0 -0.924878577539922 10 0.856160630419917 -0.897305996455662 1 1 0.737902995045191 0.223491341871422 -2.46567137030609 -0.749176972878157 11 0.562359311948542 -0.716113771919023 1 2 0.323967053996186 0.223491341871422 -1.08795977072128 -0.749176972878157 11 0.562359311948542 -0.716113771919023 1 3 0.0336190929980882 0.223491341871422 -0.110723385883767 -0.749176972878157 11 0.562359311948542 -0.716113771919023 1 4 0.223491341871422 0.223491341871422 -0.749176972878157 -0.749176972878157 11 0.562359311948542 -0.716113771919023 1 5 0 0.223491341871422 0 -0.749176972878157 11 0.562359311948542 -0.716113771919023 1 1 0.762559708748252 0.230983111600203 -1.99532721675111 -0.606917697531082 12 0.369755786024592 -0.569411772023755 1 2 0.334846651703399 0.230983111600203 -0.881905957695436 -0.606917697531082 12 0.369755786024592 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-0.354468967557447 1 2 0.350816425519658 0.241969643151537 -0.579991311488641 -0.398476506577675 14 0.160720189784965 -0.354468967557447 1 3 0.0363350035228018 0.241969643151537 -0.0575710990356399 -0.398476506577675 14 0.160720189784965 -0.354468967557447 1 4 0.241969643151537 0.241969643151537 -0.398476506577675 -0.398476506577675 14 0.160720189784965 -0.354468967557447 1 5 0 0.241969643151537 0 -0.398476506577675 14 0.160720189784965 -0.354468967557447 1 1 0.81171976023938 0.245954408217314 -1.05652281007167 -0.322968043709101 15 0.106457483619503 -0.276609372867293 1 2 0.356616338634544 0.245954408217314 -0.470621596920286 -0.322968043709101 15 0.106457483619503 -0.276609372867293 1 3 0.0369107145131582 0.245954408217314 -0.0461272730142747 -0.322968043709101 15 0.106457483619503 -0.276609372867293 1 4 0.245954408217314 0.245954408217314 -0.322968043709101 -0.322968043709101 15 0.106457483619503 -0.276609372867293 1 5 0 0.245954408217314 0 -0.322968043709101 15 0.106457483619503 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