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

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 w and b, 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 100 samples, each has 5 features, the inner join target has 5\cdot i rows where i stands for iterations. We join this with samples (100 \cdot 5 rows) based on the feature id and maximum iteration. So finally, we should have 100 \cdot 5 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 5 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	-0.569411772023755	1
3	0.0347263268569259	0.230983111600203	-0.089162358354296	-0.606917697531082	12	0.369755786024592	-0.569411772023755	1
4	0.230983111600203	0.230983111600203	-0.606917697531082	-0.606917697531082	12	0.369755786024592	-0.569411772023755	1
5	0	0.230983111600203	0	-0.606917697531082	12	0.369755786024592	-0.569411772023755	1
1	0.782512980915763	0.237052288575514	-1.61450696845941	-0.491735457602315	13	0.243492987372605	-0.450635249461157	1
2	0.343665711280353	0.237052288575514	-0.715071423930415	-0.491735457602315	13	0.243492987372605	-0.450635249461157	1
3	0.0356179504404689	0.237052288575514	-0.0717053082332894	-0.491735457602315	13	0.243492987372605	-0.450635249461157	1
4	0.237052288575514	0.237052288575514	-0.491735457602315	-0.491735457602315	13	0.243492987372605	-0.450635249461157	1
5	0	0.237052288575514	0	-0.491735457602315	13	0.243492987372605	-0.450635249461157	1
1	0.798658050600357	0.241969643151537	-1.30617096390228	-0.398476506577675	14	0.160720189784965	-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	-0.276609372867293	1
1	0.822284988340097	0.249184088654405	-0.854392070294768	-0.261831504289324	16	0.0708847130155275	-0.213572034989326	1
2	0.361322554603747	0.249184088654405	-0.382068654684519	-0.261831504289324	16	0.0708847130155275	-0.213572034989326	1
3	0.0373719872433009	0.249184088654405	-0.0368617521783987	-0.261831504289324	16	0.0708847130155275	-0.213572034989326	1
4	0.249184088654405	0.249184088654405	-0.261831504289324	-0.261831504289324	16	0.0708847130155275	-0.213572034989326	1
5	0	0.249184088654405	0	-0.261831504289324	16	0.0708847130155275	-0.213572034989326	1
1	0.830828909043044	0.251802403697298	-0.690734449052578	-0.212331387591044	17	0.0475642249141984	-0.16253573760171	1
2	0.365143241150592	0.251802403697298	-0.31037029450648	-0.212331387591044	17	0.0475642249141984	-0.16253573760171	1
3	0.0377406047650849	0.251802403697298	-0.0293599256391899	-0.212331387591044	17	0.0475642249141984	-0.16253573760171	1
4	0.251802403697298	0.251802403697298	-0.212331387591044	-0.212331387591044	17	0.0475642249141984	-0.16253573760171	1
5	0	0.251802403697298	0	-0.212331387591044	17	0.0475642249141984	-0.16253573760171	1
1	0.83773625353357	0.253925717573209	-0.558227109840943	-0.172252858358044	18	0.0322757831507861	-0.121216244655038	1
2	0.368246944095657	0.253925717573209	-0.252318475304314	-0.172252858358044	18	0.0322757831507861	-0.121216244655038	1
3	0.0380342040214768	0.253925717573209	-0.0232860980685563	-0.172252858358044	18	0.0322757831507861	-0.121216244655038	1
4	0.253925717573209	0.253925717573209	-0.172252858358044	-0.172252858358044	18	0.0322757831507861	-0.121216244655038	1
5	0	0.253925717573209	0	-0.172252858358044	18	0.0322757831507861	-0.121216244655038	1
1	0.84331852463198	0.255648246156789	-0.450941018789407	-0.139802645192794	19	0.0222527904694817	-0.0877640934587252	1
2	0.3707701288487	0.255648246156789	-0.205315763116255	-0.139802645192794	19	0.0222527904694817	-0.0877640934587252	1
3	0.0382670650021624	0.255648246156789	-0.0183684691922157	-0.139802645192794	19	0.0222527904694817	-0.0877640934587252	1
4	0.255648246156789	0.255648246156789	-0.139802645192794	-0.139802645192794	19	0.0222527904694817	-0.0877640934587252	1
5	0	0.255648246156789	0	-0.139802645192794	19	0.0222527904694817	-0.0877640934587252	1
1	0.847827934819874	0.257046272608717	-0.364075653229332	-0.113528801592203	20	0.0156815899853819	-0.0606818414675168	1
2	0.372823286479863	0.257046272608717	-0.167259110314929	-0.113528801592203	20	0.0156815899853819	-0.0606818414675168	1
3	0.0384507496940845	0.257046272608717	-0.0143869722325961	-0.113528801592203	20	0.0156815899853819	-0.0606818414675168	1
4	0.257046272608717	0.257046272608717	-0.113528801592203	-0.113528801592203	20	0.0156815899853819	-0.0606818414675168	1
5	0	0.257046272608717	0	-0.113528801592203	20	0.0156815899853819	-0.0606818414675168	1
1	0.851468691352167	0.258181560624639	-0.293744204888794	-0.0922557393821863	21	0.0113732297144537	-0.0387570640200821	1
2	0.374495877583012	0.258181560624639	-0.136445755872275	-0.0922557393821863	21	0.0113732297144537	-0.0387570640200821	1
3	0.0385946194164105	0.258181560624639	-0.0111634271392227	-0.0922557393821863	21	0.0113732297144537	-0.0387570640200821	1
4	0.258181560624639	0.258181560624639	-0.0922557393821863	-0.0922557393821863	21	0.0113732297144537	-0.0387570640200821	1
5	0	0.258181560624639	0	-0.0922557393821863	21	0.0113732297144537	-0.0387570640200821	1
1	0.854406133401055	0.259104118018461	-0.236799666746822	-0.0750316276570682	22	0.0085482861437936	-0.0210081050495532	1
2	0.375860335141735	0.259104118018461	-0.111497036233125	-0.0750316276570682	22	0.0085482861437936	-0.0210081050495532	1
3	0.0387062536878027	0.259104118018461	-0.00855356801818448	-0.0750316276570682	22	0.0085482861437936	-0.0210081050495532	1
4	0.259104118018461	0.259104118018461	-0.0750316276570682	-0.0750316276570682	22	0.0085482861437936	-0.0210081050495532	1
5	0	0.259104118018461	0	-0.0750316276570682	22	0.0085482861437936	-0.0210081050495532	1
1	0.856774130068523	0.259854434295032	-0.190694021896671	-0.0610858036175914	23	0.00669580423005917	-0.00664015341069035	1
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4	0.264281484609335	0.264281484609335	-0.00172974459308683	-0.00172974459308683	98	0.00305028338824717	0.0534725322792475	1
5	0	0.264281484609335	0	-0.00172974459308683	98	0.00305028338824717	0.0534725322792475	1
1	0.862985769694318	0.264298782055265	0.00533089778548161	-0.00172928720610077	99	0.0030486757370664	0.05345844062197	1
2	0.385533286752149	0.264298782055265	-0.00526685284024966	-0.00172928720610077	99	0.0030486757370664	0.05345844062197	1
3	0.0373447645362753	0.264298782055265	0.00249999331048842	-0.00172928720610077	99	0.0030486757370664	0.05345844062197	1
4	0.264298782055265	0.264298782055265	-0.00172928720610077	-0.00172928720610077	99	0.0030486757370664	0.05345844062197	1
5	0	0.264298782055265	0	-0.00172928720610077	99	0.0030486757370664	0.05345844062197	1
1	0.862932460716463	0.264316074927326	0.00532949678123926	-0.00172883022443804	100	0.00304706893319777	0.0534443523849859	1
2	0.385585955280552	0.264316074927326	-0.00526546295590635	-0.00172883022443804	100	0.00304706893319777	0.0534443523849859	1
3	0.0373197646031704	0.264316074927326	0.00249933459680549	-0.00172883022443804	100	0.00304706893319777	0.0534443523849859	1
4	0.264316074927326	0.264316074927326	-0.00172883022443804	-0.00172883022443804	100	0.00304706893319777	0.0534443523849859	1
5	0	0.264316074927326	0	-0.00172883022443804	100	0.00304706893319777	0.0534443523849859	1

Now you can evaluate the model.