Convexification of Learning from Constraints
Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. % For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. % In this form, the problem is resistant to standard optimization techniques. % We construct MIPs with the same solutions whose objective functions are convex. % Specifically, we characterize the tightest convex extension of the objective function, given by the Legendre-Fenchel biconjugate. % Computing values of this tightest convex extension is NP-hard. % However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently. % For some decompositions, common loss and regularization functions, we derive a closed form.
Envelope_H1_L1_3d.tex
% This file was created by matlab2tikz.
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%The latest updates can be retrieved from
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-3	0.01	3.02\\
-3	0.059	3.118\\
-3	0.108	3.216\\
-3	0.157	3.314\\
-3	0.206	3.412\\
-3	0.255	3.51\\
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-3	0.696	4.392\\
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-2.7	0.402	3.504\\
-2.7	0.451	3.602\\
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-2.7	0.598	3.896\\
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-2.7	0.794	4.288\\
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-0.3	0.5	1.3\\
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-0.3	0.794	1.888\\
-0.3	0.843	1.986\\
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-0.3	0.99	2.28\\
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0	0.451	1.098\\
0	0.5	1.00000000160102\\
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0	0.745	1.49\\
0	0.794	1.588\\
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0.3	0.108	2.084\\
0.3	0.157	1.986\\
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0.3	0.255	1.79\\
0.3	0.304	1.692\\
0.3	0.353	1.594\\
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0.3	0.451	1.398\\
0.3	0.5	1.3\\
0.3	0.549	1.202\\
0.3	0.598	1.104\\
0.3	0.647	1.006\\
0.3	0.696	1.092\\
0.3	0.745	1.19\\
0.3	0.794	1.288\\
0.3	0.843	1.386\\
0.3	0.892	1.484\\
0.3	0.941	1.582\\
0.3	0.99	1.68\\
0.6	0.01	2.58\\
0.6	0.059	2.482\\
0.6	0.108	2.384\\
0.6	0.157	2.286\\
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0.6	0.255	2.09\\
0.6	0.304	1.992\\
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0.6	0.451	1.698\\
0.6	0.5	1.6\\
0.6	0.549	1.502\\
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0.6	0.696	1.208\\
0.6	0.745	1.11\\
0.6	0.794	1.012\\
0.6	0.843	1.086\\
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0.6	0.941	1.282\\
0.6	0.99	1.38\\
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0.9	0.108	2.684\\
0.9	0.157	2.586\\
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0.9	0.255	2.39\\
0.9	0.304	2.292\\
0.9	0.353	2.194\\
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0.9	0.451	1.998\\
0.9	0.5	1.9\\
0.9	0.549	1.802\\
0.9	0.598	1.704\\
0.9	0.647	1.606\\
0.9	0.696	1.508\\
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0.9	0.843	1.214\\
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1.2	0.255	2.69\\
1.2	0.304	2.592\\
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1.2	0.549	2.102\\
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1.8	0.843	2.114\\
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1.8	0.941	1.918\\
1.8	0.99	1.82\\
2.1	0.01	4.08\\
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2.1	0.157	3.786\\
2.1	0.206	3.688\\
2.1	0.255	3.59\\
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2.1	0.353	3.394\\
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2.1	0.794	2.512\\
2.1	0.843	2.414\\
2.1	0.892	2.316\\
2.1	0.941	2.218\\
2.1	0.99	2.12\\
2.4	0.01	4.38\\
2.4	0.059	4.282\\
2.4	0.108	4.184\\
2.4	0.157	4.086\\
2.4	0.206	3.988\\
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2.7	0.647	3.406\\
2.7	0.696	3.308\\
2.7	0.745	3.21\\
2.7	0.794	3.112\\
2.7	0.843	3.014\\
2.7	0.892	2.916\\
2.7	0.941	2.818\\
2.7	0.99	2.72\\
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3	0.402	4.196\\
3	0.451	4.098\\
3	0.5	4\\
3	0.549	3.902\\
3	0.598	3.804\\
3	0.647	3.706\\
3	0.696	3.608\\
3	0.745	3.51\\
3	0.794	3.412\\
3	0.843	3.314\\
3	0.892	3.216\\
3	0.941	3.118\\
3	0.99	3.02\\
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-2.4	0	2.4\\
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0.3	0	2.9\\
0.6	0	3.8\\
0.9	0	4.7\\
1.2	0	5.6\\
1.5	0	6.5\\
1.8	0	7.4\\
2.1	0	8.3\\
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-0.3	1	2.9\\
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0.3	1	1.7\\
0.6	1	1.4\\
0.9	1	1.1\\
1.2	1	1.2\\
1.5	1	1.5\\
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2.1	1	2.1\\
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2.7	1	2.7\\
3	1	3\\
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Envelope_H1_L1_BND_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
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Envelope_H1_L1_ORG_3d.tex
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-2.7	0.99	9.992\\
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0	0.01	2\\
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0	0.206	2\\
0	0.255	2\\
0	0.304	2\\
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0	0.402	2\\
0	0.451	2\\
0	0.5	2\\
0	0.549	2\\
0	0.598	2\\
0	0.647	2\\
0	0.696	2\\
0	0.745	2\\
0	0.794	2\\
0	0.843	2\\
0	0.892	2\\
0	0.941	2\\
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0.3	0.402	2.4176\\
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0.3	0.696	2.0648\\
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0.3	0.794	1.9472\\
0.3	0.843	1.8884\\
0.3	0.892	1.8296\\
0.3	0.941	1.7708\\
0.3	0.99	1.712\\
0.6	0.01	3.776\\
0.6	0.059	3.6584\\
0.6	0.108	3.5408\\
0.6	0.157	3.4232\\
0.6	0.206	3.3056\\
0.6	0.255	3.188\\
0.6	0.304	3.0704\\
0.6	0.353	2.9528\\
0.6	0.402	2.8352\\
0.6	0.451	2.7176\\
0.6	0.5	2.6\\
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0.6	0.598	2.3648\\
0.6	0.647	2.2472\\
0.6	0.696	2.1296\\
0.6	0.745	2.012\\
0.6	0.794	1.8944\\
0.6	0.843	1.7768\\
0.6	0.892	1.6592\\
0.6	0.941	1.5416\\
0.6	0.99	1.424\\
0.9	0.01	4.664\\
0.9	0.059	4.4876\\
0.9	0.108	4.3112\\
0.9	0.157	4.1348\\
0.9	0.206	3.9584\\
0.9	0.255	3.782\\
0.9	0.304	3.6056\\
0.9	0.353	3.4292\\
0.9	0.402	3.2528\\
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0.9	0.5	2.9\\
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0.9	0.598	2.5472\\
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0.9	0.696	2.1944\\
0.9	0.745	2.018\\
0.9	0.794	1.8416\\
0.9	0.843	1.6652\\
0.9	0.892	1.4888\\
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1.2	0.255	4.376\\
1.2	0.304	4.1408\\
1.2	0.353	3.9056\\
1.2	0.402	3.6704\\
1.2	0.451	3.4352\\
1.2	0.5	3.2\\
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1.2	0.598	2.7296\\
1.2	0.647	2.4944\\
1.2	0.696	2.2592\\
1.2	0.745	2.024\\
1.2	0.794	1.7888\\
1.2	0.843	1.5536\\
1.2	0.892	1.3184\\
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1.5	0.01	6.44\\
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1.5	0.402	4.088\\
1.5	0.451	3.794\\
1.5	0.5	3.5\\
1.5	0.549	3.206\\
1.5	0.598	2.912\\
1.5	0.647	2.618\\
1.5	0.696	2.324\\
1.5	0.745	2.03\\
1.5	0.794	1.736\\
1.5	0.843	1.5\\
1.5	0.892	1.5\\
1.5	0.941	1.5\\
1.5	0.99	1.5\\
1.8	0.01	7.328\\
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1.8	0.157	6.2696\\
1.8	0.206	5.9168\\
1.8	0.255	5.564\\
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1.8	0.353	4.8584\\
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1.8	0.5	3.8\\
1.8	0.549	3.4472\\
1.8	0.598	3.0944\\
1.8	0.647	2.7416\\
1.8	0.696	2.3888\\
1.8	0.745	2.036\\
1.8	0.794	1.8\\
1.8	0.843	1.8\\
1.8	0.892	1.8\\
1.8	0.941	1.8\\
1.8	0.99	1.8\\
2.1	0.01	8.216\\
2.1	0.059	7.8044\\
2.1	0.108	7.3928\\
2.1	0.157	6.9812\\
2.1	0.206	6.5696\\
2.1	0.255	6.158\\
2.1	0.304	5.7464\\
2.1	0.353	5.3348\\
2.1	0.402	4.9232\\
2.1	0.451	4.5116\\
2.1	0.5	4.1\\
2.1	0.549	3.6884\\
2.1	0.598	3.2768\\
2.1	0.647	2.8652\\
2.1	0.696	2.4536\\
2.1	0.745	2.1\\
2.1	0.794	2.1\\
2.1	0.843	2.1\\
2.1	0.892	2.1\\
2.1	0.941	2.1\\
2.1	0.99	2.1\\
2.4	0.01	9.104\\
2.4	0.059	8.6336\\
2.4	0.108	8.1632\\
2.4	0.157	7.6928\\
2.4	0.206	7.2224\\
2.4	0.255	6.752\\
2.4	0.304	6.2816\\
2.4	0.353	5.8112\\
2.4	0.402	5.3408\\
2.4	0.451	4.8704\\
2.4	0.5	4.4\\
2.4	0.549	3.9296\\
2.4	0.598	3.4592\\
2.4	0.647	2.9888\\
2.4	0.696	2.5184\\
2.4	0.745	2.4\\
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2.4	0.892	2.4\\
2.4	0.941	2.4\\
2.4	0.99	2.4\\
2.7	0.01	9.992\\
2.7	0.059	9.4628\\
2.7	0.108	8.9336\\
2.7	0.157	8.4044\\
2.7	0.206	7.8752\\
2.7	0.255	7.346\\
2.7	0.304	6.8168\\
2.7	0.353	6.2876\\
2.7	0.402	5.7584\\
2.7	0.451	5.2292\\
2.7	0.5	4.7\\
2.7	0.549	4.1708\\
2.7	0.598	3.6416\\
2.7	0.647	3.1124\\
2.7	0.696	2.7\\
2.7	0.745	2.7\\
2.7	0.794	2.7\\
2.7	0.843	2.7\\
2.7	0.892	2.7\\
2.7	0.941	2.7\\
2.7	0.99	2.7\\
3	0.01	10.88\\
3	0.059	10.292\\
3	0.108	9.704\\
3	0.157	9.116\\
3	0.206	8.528\\
3	0.255	7.94\\
3	0.304	7.352\\
3	0.353	6.764\\
3	0.402	6.176\\
3	0.451	5.588\\
3	0.5	5\\
3	0.549	4.412\\
3	0.598	3.824\\
3	0.647	3.236\\
3	0.696	3\\
3	0.745	3\\
3	0.794	3\\
3	0.843	3\\
3	0.892	3\\
3	0.941	3\\
3	0.99	3\\
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0.3	0	2.9\\
0.6	0	3.8\\
0.9	0	4.7\\
1.2	0	5.6\\
1.5	0	6.5\\
1.8	0	7.4\\
2.1	0	8.3\\
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};

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0	1	2\\
0.3	1	1.7\\
0.6	1	1.4\\
0.9	1	1.1\\
1.2	1	1.2\\
1.5	1	1.5\\
1.8	1	1.8\\
2.1	1	2.1\\
2.4	1	2.4\\
2.7	1	2.7\\
3	1	3\\
};
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ext_cvx.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}

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    width=0.33\textwidth,
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1.5	0	13.625\\
1.8	0	15.62\\
2.1	0	17.705\\
2.4	0	19.88\\
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3	0	24.5\\
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1.2	1	0.72\\
1.5	1	1.125\\
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2.1	1	2.205\\
2.4	1	2.88\\
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3	1	4.5\\
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table[
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-3	0.698	15.82505\\
-3	0.7475	17.090703125\\
-3	0.797	18.4176125\\
-3	0.8465	19.805778125\\
-3	0.896	21.2552\\
-3	0.9455	22.765878125\\
-3	0.995	24.3378125\\
-2.7	0.005	3.67939698492462\\
-2.7	0.0545	4.03955579058699\\
-2.7	0.104	4.43950892857143\\
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ext_ncvx.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}

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-2.7	1	22.145\\
-2.4	1	19.88\\
-2.1	1	17.705\\
-1.8	1	15.62\\
-1.5	1	13.625\\
-1.2	1	11.72\\
-0.9	1	9.905\\
-0.6	1	8.18\\
-0.3	1	6.545\\
0	1	5\\
0.3	1	3.545\\
0.6	1	2.18\\
0.9	1	0.905\\
1.2	1	0.72\\
1.5	1	1.125\\
1.8	1	1.62\\
2.1	1	2.205\\
2.4	1	2.88\\
2.7	1	3.645\\
3	1	4.5\\
};

\addplot3[%
    surf,
    shader=faceted,
    mesh/rows=21
]
table[
    row sep=crcr,
    header=false
] {%
%
-3	0.005	4.5\\
-3	0.0545	4.5\\
-3	0.104	4.5\\
-3	0.1535	4.5\\
-3	0.203	4.5\\
-3	0.2525	4.5\\
-3	0.302	4.5\\
-3	0.3515	5.045\\
-3	0.401	6.53\\
-3	0.4505	8.015\\
-3	0.5	9.5\\
-3	0.5495	10.985\\
-3	0.599	12.47\\
-3	0.6485	13.955\\
-3	0.698	15.44\\
-3	0.7475	16.925\\
-3	0.797	18.41\\
-3	0.8465	19.895\\
-3	0.896	21.38\\
-3	0.9455	22.865\\
-3	0.995	24.35\\
-2.7	0.005	3.645\\
-2.7	0.0545	3.645\\
-2.7	0.104	3.645\\
-2.7	0.1535	3.645\\
-2.7	0.203	3.645\\
-2.7	0.2525	3.645\\
-2.7	0.302	3.645\\
-2.7	0.3515	4.6355\\
-2.7	0.401	5.972\\
-2.7	0.4505	7.3085\\
-2.7	0.5	8.645\\
-2.7	0.5495	9.9815\\
-2.7	0.599	11.318\\
-2.7	0.6485	12.6545\\
-2.7	0.698	13.991\\
-2.7	0.7475	15.3275\\
-2.7	0.797	16.664\\
-2.7	0.8465	18.0005\\
-2.7	0.896	19.337\\
-2.7	0.9455	20.6735\\
-2.7	0.995	22.01\\
-2.4	0.005	2.88\\
-2.4	0.0545	2.88\\
-2.4	0.104	2.88\\
-2.4	0.1535	2.88\\
-2.4	0.203	2.88\\
-2.4	0.2525	2.88\\
-2.4	0.302	3.128\\
-2.4	0.3515	4.316\\
-2.4	0.401	5.504\\
-2.4	0.4505	6.692\\
-2.4	0.5	7.88\\
-2.4	0.5495	9.068\\
-2.4	0.599	10.256\\
-2.4	0.6485	11.444\\
-2.4	0.698	12.632\\
-2.4	0.7475	13.82\\
-2.4	0.797	15.008\\
-2.4	0.8465	16.196\\
-2.4	0.896	17.384\\
-2.4	0.9455	18.572\\
-2.4	0.995	19.76\\
-2.1	0.005	2.205\\
-2.1	0.0545	2.205\\
-2.1	0.104	2.205\\
-2.1	0.1535	2.205\\
-2.1	0.203	2.205\\
-2.1	0.2525	2.205\\
-2.1	0.302	3.047\\
-2.1	0.3515	4.0865\\
-2.1	0.401	5.126\\
-2.1	0.4505	6.1655\\
-2.1	0.5	7.205\\
-2.1	0.5495	8.2445\\
-2.1	0.599	9.284\\
-2.1	0.6485	10.3235\\
-2.1	0.698	11.363\\
-2.1	0.7475	12.4025\\
-2.1	0.797	13.442\\
-2.1	0.8465	14.4815\\
-2.1	0.896	15.521\\
-2.1	0.9455	16.5605\\
-2.1	0.995	17.6\\
-1.8	0.005	1.62\\
-1.8	0.0545	1.62\\
-1.8	0.104	1.62\\
-1.8	0.1535	1.62\\
-1.8	0.203	1.62\\
-1.8	0.2525	2.165\\
-1.8	0.302	3.056\\
-1.8	0.3515	3.947\\
-1.8	0.401	4.838\\
-1.8	0.4505	5.729\\
-1.8	0.5	6.62\\
-1.8	0.5495	7.511\\
-1.8	0.599	8.402\\
-1.8	0.6485	9.293\\
-1.8	0.698	10.184\\
-1.8	0.7475	11.075\\
-1.8	0.797	11.966\\
-1.8	0.8465	12.857\\
-1.8	0.896	13.748\\
-1.8	0.9455	14.639\\
-1.8	0.995	15.53\\
-1.5	0.005	1.125\\
-1.5	0.0545	1.125\\
-1.5	0.104	1.125\\
-1.5	0.1535	1.125\\
-1.5	0.203	1.67\\
-1.5	0.2525	2.4125\\
-1.5	0.302	3.155\\
-1.5	0.3515	3.8975\\
-1.5	0.401	4.64\\
-1.5	0.4505	5.3825\\
-1.5	0.5	6.125\\
-1.5	0.5495	6.8675\\
-1.5	0.599	7.61\\
-1.5	0.6485	8.3525\\
-1.5	0.698	9.095\\
-1.5	0.7475	9.8375\\
-1.5	0.797	10.58\\
-1.5	0.8465	11.3225\\
-1.5	0.896	12.065\\
-1.5	0.9455	12.8075\\
-1.5	0.995	13.55\\
-1.2	0.005	0.72\\
-1.2	0.0545	0.72\\
-1.2	0.104	0.968\\
-1.2	0.1535	1.562\\
-1.2	0.203	2.156\\
-1.2	0.2525	2.75\\
-1.2	0.302	3.344\\
-1.2	0.3515	3.938\\
-1.2	0.401	4.532\\
-1.2	0.4505	5.126\\
-1.2	0.5	5.72\\
-1.2	0.5495	6.314\\
-1.2	0.599	6.908\\
-1.2	0.6485	7.502\\
-1.2	0.698	8.096\\
-1.2	0.7475	8.69\\
-1.2	0.797	9.284\\
-1.2	0.8465	9.878\\
-1.2	0.896	10.472\\
-1.2	0.9455	11.066\\
-1.2	0.995	11.66\\
-0.9	0.005	0.95\\
-0.9	0.0545	1.3955\\
-0.9	0.104	1.841\\
-0.9	0.1535	2.2865\\
-0.9	0.203	2.732\\
-0.9	0.2525	3.1775\\
-0.9	0.302	3.623\\
-0.9	0.3515	4.0685\\
-0.9	0.401	4.514\\
-0.9	0.4505	4.9595\\
-0.9	0.5	5.405\\
-0.9	0.5495	5.8505\\
-0.9	0.599	6.296\\
-0.9	0.6485	6.7415\\
-0.9	0.698	7.187\\
-0.9	0.7475	7.6325\\
-0.9	0.797	8.078\\
-0.9	0.8465	8.5235\\
-0.9	0.896	8.969\\
-0.9	0.9455	9.4145\\
-0.9	0.995	9.86\\
-0.6	0.005	2.21\\
-0.6	0.0545	2.507\\
-0.6	0.104	2.804\\
-0.6	0.1535	3.101\\
-0.6	0.203	3.398\\
-0.6	0.2525	3.695\\
-0.6	0.302	3.992\\
-0.6	0.3515	4.289\\
-0.6	0.401	4.586\\
-0.6	0.4505	4.883\\
-0.6	0.5	5.18\\
-0.6	0.5495	5.477\\
-0.6	0.599	5.774\\
-0.6	0.6485	6.071\\
-0.6	0.698	6.368\\
-0.6	0.7475	6.665\\
-0.6	0.797	6.962\\
-0.6	0.8465	7.259\\
-0.6	0.896	7.556\\
-0.6	0.9455	7.853\\
-0.6	0.995	8.15\\
-0.3	0.005	3.56\\
-0.3	0.0545	3.7085\\
-0.3	0.104	3.857\\
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-0.3	0.3515	4.5995\\
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-0.3	0.4505	4.8965\\
-0.3	0.5	5.045\\
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-0.3	0.599	5.342\\
-0.3	0.6485	5.4905\\
-0.3	0.698	5.639\\
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-0.3	0.8465	6.0845\\
-0.3	0.896	6.233\\
-0.3	0.9455	6.3815\\
-0.3	0.995	6.53\\
0	0.005	5\\
0	0.0545	5\\
0	0.104	5\\
0	0.1535	5\\
0	0.203	5\\
0	0.2525	5\\
0	0.302	5\\
0	0.3515	5\\
0	0.401	5\\
0	0.4505	5\\
0	0.5	5\\
0	0.5495	5\\
0	0.599	5\\
0	0.6485	5\\
0	0.698	5\\
0	0.7475	5\\
0	0.797	5\\
0	0.8465	5\\
0	0.896	5\\
0	0.9455	5\\
0	0.995	5\\
0.3	0.005	6.53\\
0.3	0.0545	6.3815\\
0.3	0.104	6.233\\
0.3	0.1535	6.0845\\
0.3	0.203	5.936\\
0.3	0.2525	5.7875\\
0.3	0.302	5.639\\
0.3	0.3515	5.4905\\
0.3	0.401	5.342\\
0.3	0.4505	5.1935\\
0.3	0.5	5.045\\
0.3	0.5495	4.8965\\
0.3	0.599	4.748\\
0.3	0.6485	4.5995\\
0.3	0.698	4.451\\
0.3	0.7475	4.3025\\
0.3	0.797	4.154\\
0.3	0.8465	4.0055\\
0.3	0.896	3.857\\
0.3	0.9455	3.7085\\
0.3	0.995	3.56\\
0.6	0.005	8.15\\
0.6	0.0545	7.853\\
0.6	0.104	7.556\\
0.6	0.1535	7.259\\
0.6	0.203	6.962\\
0.6	0.2525	6.665\\
0.6	0.302	6.368\\
0.6	0.3515	6.071\\
0.6	0.401	5.774\\
0.6	0.4505	5.477\\
0.6	0.5	5.18\\
0.6	0.5495	4.883\\
0.6	0.599	4.586\\
0.6	0.6485	4.289\\
0.6	0.698	3.992\\
0.6	0.7475	3.695\\
0.6	0.797	3.398\\
0.6	0.8465	3.101\\
0.6	0.896	2.804\\
0.6	0.9455	2.507\\
0.6	0.995	2.21\\
0.9	0.005	9.86\\
0.9	0.0545	9.4145\\
0.9	0.104	8.969\\
0.9	0.1535	8.5235\\
0.9	0.203	8.078\\
0.9	0.2525	7.6325\\
0.9	0.302	7.187\\
0.9	0.3515	6.7415\\
0.9	0.401	6.296\\
0.9	0.4505	5.8505\\
0.9	0.5	5.405\\
0.9	0.5495	4.9595\\
0.9	0.599	4.514\\
0.9	0.6485	4.0685\\
0.9	0.698	3.623\\
0.9	0.7475	3.1775\\
0.9	0.797	2.732\\
0.9	0.8465	2.2865\\
0.9	0.896	1.841\\
0.9	0.9455	1.3955\\
0.9	0.995	0.95\\
1.2	0.005	11.66\\
1.2	0.0545	11.066\\
1.2	0.104	10.472\\
1.2	0.1535	9.878\\
1.2	0.203	9.284\\
1.2	0.2525	8.69\\
1.2	0.302	8.096\\
1.2	0.3515	7.502\\
1.2	0.401	6.908\\
1.2	0.4505	6.314\\
1.2	0.5	5.72\\
1.2	0.5495	5.126\\
1.2	0.599	4.532\\
1.2	0.6485	3.938\\
1.2	0.698	3.344\\
1.2	0.7475	2.75\\
1.2	0.797	2.156\\
1.2	0.8465	1.562\\
1.2	0.896	0.968\\
1.2	0.9455	0.72\\
1.2	0.995	0.72\\
1.5	0.005	13.55\\
1.5	0.0545	12.8075\\
1.5	0.104	12.065\\
1.5	0.1535	11.3225\\
1.5	0.203	10.58\\
1.5	0.2525	9.8375\\
1.5	0.302	9.095\\
1.5	0.3515	8.3525\\
1.5	0.401	7.61\\
1.5	0.4505	6.8675\\
1.5	0.5	6.125\\
1.5	0.5495	5.3825\\
1.5	0.599	4.64\\
1.5	0.6485	3.8975\\
1.5	0.698	3.155\\
1.5	0.7475	2.4125\\
1.5	0.797	1.67\\
1.5	0.8465	1.125\\
1.5	0.896	1.125\\
1.5	0.9455	1.125\\
1.5	0.995	1.125\\
1.8	0.005	15.53\\
1.8	0.0545	14.639\\
1.8	0.104	13.748\\
1.8	0.1535	12.857\\
1.8	0.203	11.966\\
1.8	0.2525	11.075\\
1.8	0.302	10.184\\
1.8	0.3515	9.293\\
1.8	0.401	8.402\\
1.8	0.4505	7.511\\
1.8	0.5	6.62\\
1.8	0.5495	5.729\\
1.8	0.599	4.838\\
1.8	0.6485	3.947\\
1.8	0.698	3.056\\
1.8	0.7475	2.165\\
1.8	0.797	1.62\\
1.8	0.8465	1.62\\
1.8	0.896	1.62\\
1.8	0.9455	1.62\\
1.8	0.995	1.62\\
2.1	0.005	17.6\\
2.1	0.0545	16.5605\\
2.1	0.104	15.521\\
2.1	0.1535	14.4815\\
2.1	0.203	13.442\\
2.1	0.2525	12.4025\\
2.1	0.302	11.363\\
2.1	0.3515	10.3235\\
2.1	0.401	9.284\\
2.1	0.4505	8.2445\\
2.1	0.5	7.205\\
2.1	0.5495	6.1655\\
2.1	0.599	5.126\\
2.1	0.6485	4.0865\\
2.1	0.698	3.047\\
2.1	0.7475	2.205\\
2.1	0.797	2.205\\
2.1	0.8465	2.205\\
2.1	0.896	2.205\\
2.1	0.9455	2.205\\
2.1	0.995	2.205\\
2.4	0.005	19.76\\
2.4	0.0545	18.572\\
2.4	0.104	17.384\\
2.4	0.1535	16.196\\
2.4	0.203	15.008\\
2.4	0.2525	13.82\\
2.4	0.302	12.632\\
2.4	0.3515	11.444\\
2.4	0.401	10.256\\
2.4	0.4505	9.068\\
2.4	0.5	7.88\\
2.4	0.5495	6.692\\
2.4	0.599	5.504\\
2.4	0.6485	4.316\\
2.4	0.698	3.128\\
2.4	0.7475	2.88\\
2.4	0.797	2.88\\
2.4	0.8465	2.88\\
2.4	0.896	2.88\\
2.4	0.9455	2.88\\
2.4	0.995	2.88\\
2.7	0.005	22.01\\
2.7	0.0545	20.6735\\
2.7	0.104	19.337\\
2.7	0.1535	18.0005\\
2.7	0.203	16.664\\
2.7	0.2525	15.3275\\
2.7	0.302	13.991\\
2.7	0.3515	12.6545\\
2.7	0.401	11.318\\
2.7	0.4505	9.9815\\
2.7	0.5	8.645\\
2.7	0.5495	7.3085\\
2.7	0.599	5.972\\
2.7	0.6485	4.6355\\
2.7	0.698	3.645\\
2.7	0.7475	3.645\\
2.7	0.797	3.645\\
2.7	0.8465	3.645\\
2.7	0.896	3.645\\
2.7	0.9455	3.645\\
2.7	0.995	3.645\\
3	0.005	24.35\\
3	0.0545	22.865\\
3	0.104	21.38\\
3	0.1535	19.895\\
3	0.203	18.41\\
3	0.2525	16.925\\
3	0.302	15.44\\
3	0.3515	13.955\\
3	0.401	12.47\\
3	0.4505	10.985\\
3	0.5	9.5\\
3	0.5495	8.015\\
3	0.599	6.53\\
3	0.6485	5.045\\
3	0.698	4.5\\
3	0.7475	4.5\\
3	0.797	4.5\\
3	0.8465	4.5\\
3	0.896	4.5\\
3	0.9455	4.5\\
3	0.995	4.5\\
};

 \end{axis}
\end{tikzpicture}%

ext_triv.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}
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Envelope_H1_L1_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
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%where you can also make suggestions and rate matlab2tikz.
%
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Envelope_H1_L1_BND_3d.tex
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% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
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%where you can also make suggestions and rate matlab2tikz.
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Envelope_H1_L2_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
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Envelope_H2_L1_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}

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width=\smallplotwidth,
every outer x axis line/.append style={black},
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Envelope_H2_L1_BND_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
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};
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Envelope_H2_L2_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}

\begin{axis}[%
width=\smallplotwidth,
every outer x axis line/.append style={black},
every x tick label/.append style={font=\color{black}},
xmin=-3,
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 table[row sep=crcr] {%
-3	0	9\\
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1.8	0	34.6\\
2.1	0	42.85\\
2.4	0	52\\
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3	0	73\\
};
 \addplot3 [solid]
 table[row sep=crcr] {%
-3	1	73\\
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0	1	4\\
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0.6	1	1\\
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1.2	1	1.44\\
1.5	1	2.25\\
1.8	1	3.24\\
2.1	1	4.41\\
2.4	1	5.76\\
2.7	1	7.29\\
3	1	9\\
};
 \end{axis}
\end{tikzpicture}%

Envelope_Logistic_L1_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}

\begin{axis}[%
width=\smallplotwidth,
every outer x axis line/.append style={black},
every x tick label/.append style={font=\color{black}},
xmin=-3,
xmax=3,
tick align=outside,
xlabel={$\theta$},
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xmajorgrids,
every outer y axis line/.append style={black},
every y tick label/.append style={font=\color{black}},
ymin=0,
ymax=1,
ylabel={y},
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ymajorgrids,
every outer z axis line/.append style={black},
every z tick label/.append style={font=\color{black}},
zmin=2.38629436111989,
zmax=11.0362998558356,
zmajorgrids,
view={-18.5}{34},
axis x line*=bottom,
axis y line*=left,
axis z line*=left,
colormap={mycolormap}{color(0)=(white) color(1)=(mycolor1)}
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Envelope_Logistic_L1_BND_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
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%where you can also make suggestions and rate matlab2tikz.
%
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width=\smallplotwidth,
every outer x axis line/.append style={black},
every x tick label/.append style={font=\color{black}},
xmin=-3,
xmax=3,
tick align=outside,
xlabel={$\theta$},
    xlabel style={yshift=3ex},
xmajorgrids,
every outer y axis line/.append style={black},
every y tick label/.append style={font=\color{black}},
ymin=0,
ymax=1,
ylabel={y},
	ylabel style={yshift=3ex},
ymajorgrids,
every outer z axis line/.append style={black},
every z tick label/.append style={font=\color{black}},
zmin=2.38629436111989,
zmax=11.0362998558356,
zmajorgrids,
view={-18.5}{34},
axis x line*=bottom,
axis y line*=left,
axis z line*=left,
colormap={mycolormap}{color(0)=(white) color(1)=(mycolor1)}
]

\addplot3[%
    surf,
    shader=faceted,
    mesh/rows=21
]
table[row sep=crcr,header=false] {%
%
-3	0.01	3.27052523744491\\
-3	0.059	3.51140464113207\\
-3	0.108	3.90053885104465\\
-3	0.157	4.29185375712209\\
-3	0.206	4.68359296940036\\
-3	0.255	5.07548362131038\\
-3	0.304	5.46744533289545\\
-3	0.353	5.85944588017059\\
-3	0.402	6.25146995958176\\
-3	0.451	6.6435093666286\\
-3	0.5	7.03555931045843\\
-3	0.549	7.42761680695385\\
-3	0.598	7.81967990092136\\
-3	0.647	8.21174725805037\\
-3	0.696	8.60381793669924\\
-3	0.745	8.99589342764152\\
-3	0.794	9.38796887890656\\
-3	0.843	9.78004607738151\\
-3	0.892	10.1721247292636\\
-3	0.941	10.5642046185763\\
-3	0.99	10.9562841879095\\
-2.7	0.01	3.05023601804184\\
-2.7	0.059	3.12491224755173\\
-2.7	0.108	3.20444969292106\\
-2.7	0.157	3.44248109853084\\
-2.7	0.206	3.8176876612142\\
-2.7	0.255	4.20216158256286\\
-2.7	0.304	4.59004958140428\\
-2.7	0.353	4.9795124140841\\
-2.7	0.402	5.36981678074714\\
-2.7	0.451	5.76061910036982\\
-2.7	0.5	6.15173888217369\\
-2.7	0.549	6.54307260578232\\
-2.7	0.598	6.93455716238009\\
-2.7	0.647	7.32615177178253\\
-2.7	0.696	7.71782924511804\\
-2.7	0.745	8.10957250888609\\
-2.7	0.794	8.50136384708369\\
-2.7	0.843	8.89319527228838\\
-2.7	0.892	9.28505925963142\\
-2.7	0.941	9.67695006921301\\
-2.7	0.99	10.0688639024132\\
-2.4	0.01	2.85328840712814\\
-2.4	0.059	2.9170480407981\\
-2.4	0.108	2.98560017114325\\
-2.4	0.157	3.05927373101702\\
-2.4	0.206	3.13837678944409\\
-2.4	0.255	3.38016980580721\\
-2.4	0.304	3.74684080160606\\
-2.4	0.353	4.12433977290931\\
-2.4	0.402	4.50719184412834\\
-2.4	0.451	4.89299347166476\\
-2.4	0.5	5.28056310495942\\
-2.4	0.549	5.66926432969026\\
-2.4	0.598	6.05872837412375\\
-2.4	0.647	6.44872865731915\\
-2.4	0.696	6.83911882151694\\
-2.4	0.745	7.2298027385296\\
-2.4	0.794	7.62070838853988\\
-2.4	0.843	8.01178902020807\\
-2.4	0.892	8.40300898471026\\
-2.4	0.941	8.79434162424824\\
-2.4	0.99	9.18576694045089\\
-2.1	0.01	2.68478327091351\\
-2.1	0.059	2.73683348955113\\
-2.1	0.108	2.79332984537706\\
-2.1	0.157	2.85469331011348\\
-2.1	0.206	2.92135877995937\\
-2.1	0.255	2.99376015250217\\
-2.1	0.304	3.0723076551846\\
-2.1	0.353	3.31926743506196\\
-2.1	0.402	3.68074641711836\\
-2.1	0.451	4.05281233790443\\
-2.1	0.5	4.43103906636244\\
-2.1	0.549	4.81307918048496\\
-2.1	0.598	5.19760728976896\\
-2.1	0.647	5.58383190992735\\
-2.1	0.696	5.97125692560677\\
-2.1	0.745	6.35956019380564\\
-2.1	0.794	6.7485182604534\\
-2.1	0.843	7.13797991178342\\
-2.1	0.892	7.52783533867898\\
-2.1	0.941	7.91800398623939\\
-2.1	0.99	8.30842592213787\\
-1.8	0.01	2.54988565748489\\
-1.8	0.059	2.58996834469786\\
-1.8	0.108	2.63397352011845\\
-1.8	0.157	2.68217188167963\\
-1.8	0.206	2.73515390119583\\
-1.8	0.255	2.79346838722306\\
-1.8	0.304	2.85769674753758\\
-1.8	0.353	2.9284321493685\\
-1.8	0.402	3.00624416751658\\
-1.8	0.451	3.25888847361394\\
-1.8	0.5	3.61693282116483\\
-1.8	0.549	3.98486366117059\\
-1.8	0.598	4.35917065913558\\
-1.8	0.647	4.73775364935253\\
-1.8	0.696	5.11930731479876\\
-1.8	0.745	5.5029917377332\\
-1.8	0.794	5.88824207567428\\
-1.8	0.843	6.27467433632779\\
-1.8	0.892	6.66201680320954\\
-1.8	0.941	7.05007279310507\\
-1.8	0.99	7.43869777363205\\
-1.5	0.01	2.45314618832928\\
-1.5	0.059	2.48029352378435\\
-1.5	0.108	2.51181076275321\\
-1.5	0.157	2.54756246913536\\
-1.5	0.206	2.5874000795498\\
-1.5	0.255	2.63117088283529\\
-1.5	0.304	2.67964715451021\\
-1.5	0.353	2.73396726988565\\
-1.5	0.402	2.79493464618646\\
-1.5	0.451	2.8633970108452\\
-1.5	0.5	2.94018969856472\\
-1.5	0.549	3.19876036247516\\
-1.5	0.598	3.5543769943811\\
-1.5	0.647	3.91911076572969\\
-1.5	0.696	4.29015387414881\\
-1.5	0.745	4.66568289837827\\
-1.5	0.794	5.04447516697117\\
-1.5	0.843	5.42570486354856\\
-1.5	0.892	5.80878971039178\\
-1.5	0.941	6.19330894479694\\
-1.5	0.99	6.57896023672669\\
-1.2	0.01	2.39837003790125\\
-1.2	0.059	2.41146955228516\\
-1.2	0.108	2.42924995135038\\
-1.2	0.157	2.45162883740865\\
-1.2	0.206	2.47850455871586\\
-1.2	0.255	2.5097579995812\\
-1.2	0.304	2.54525459992265\\
-1.2	0.353	2.58484653525922\\
-1.2	0.402	2.62837709876171\\
-1.2	0.451	2.6772769689797\\
-1.2	0.5	2.73367909743919\\
-1.2	0.549	2.79886395059446\\
-1.2	0.598	2.87415097231992\\
-1.2	0.647	3.13877169107037\\
-1.2	0.696	3.49258530912163\\
-1.2	0.745	3.85479246580019\\
-1.2	0.794	4.22311542799841\\
-1.2	0.843	4.59597357976371\\
-1.2	0.892	4.97226830928679\\
-1.2	0.941	5.35120225870314\\
-1.2	0.99	5.73220130346654\\
-0.9	0.01	2.38789393463518\\
-0.9	0.059	2.38637536002642\\
-0.9	0.108	2.38965647672745\\
-0.9	0.157	2.39772158098789\\
-0.9	0.206	2.41053229017634\\
-0.9	0.255	2.42802826818129\\
-0.9	0.304	2.45012834804117\\
-0.9	0.353	2.47673200807131\\
-0.9	0.402	2.50772114576166\\
-0.9	0.451	2.54296208524204\\
-0.9	0.5	2.58230774946426\\
-0.9	0.549	2.6255999274119\\
-0.9	0.598	2.67522499879683\\
-0.9	0.647	2.73530154345735\\
-0.9	0.696	2.8081436044403\\
-0.9	0.745	3.07886848847821\\
-0.9	0.794	3.43129168295551\\
-0.9	0.843	3.79145971650822\\
-0.9	0.892	4.15749426888951\\
-0.9	0.941	4.52803178654287\\
-0.9	0.99	4.90207912144332\\
-0.6	0.01	2.42217922773193\\
-0.6	0.059	2.40610983257865\\
-0.6	0.108	2.39474644811729\\
-0.6	0.157	2.38814279160052\\
-0.6	0.206	2.3863303609039\\
-0.6	0.255	2.38931783724485\\
-0.6	0.304	2.39709091952962\\
-0.6	0.353	2.40961259700518\\
-0.6	0.402	2.42682384923882\\
-0.6	0.451	2.44864474554626\\
-0.6	0.5	2.47497590097178\\
-0.6	0.549	2.50570023373252\\
-0.6	0.598	2.54068496033425\\
-0.6	0.647	2.57978375968098\\
-0.6	0.696	2.62283903643636\\
-0.6	0.745	2.67401084067003\\
-0.6	0.794	2.74221404265457\\
-0.6	0.843	3.01902144482984\\
-0.6	0.892	3.3703404783517\\
-0.6	0.941	3.72883365100419\\
-0.6	0.99	4.09292423041287\\
-0.3	0.01	2.49973346992733\\
-0.3	0.059	2.46980652885517\\
-0.3	0.108	2.44429550876825\\
-0.3	0.157	2.4233143844574\\
-0.3	0.206	2.40695909129721\\
-0.3	0.255	2.39530581859723\\
-0.3	0.304	2.38840961529803\\
-0.3	0.353	2.38630336110639\\
-0.3	0.402	2.38899714339521\\
-0.3	0.451	2.39647806475279\\
-0.3	0.5	2.40871048893706\\
-0.3	0.549	2.42563671532875\\
-0.3	0.598	2.44717805499582\\
-0.3	0.647	2.47323626631913\\
-0.3	0.696	2.50369529574317\\
-0.3	0.745	2.5384232602835\\
-0.3	0.794	2.57727460330158\\
-0.3	0.843	2.62009235376408\\
-0.3	0.892	2.67658942627176\\
-0.3	0.941	2.95921536609258\\
-0.3	0.99	3.30963438692057\\
0	0.01	2.61735991847432\\
0	0.059	2.57478031759191\\
0	0.108	2.53617702001556\\
0	0.157	2.50170636385257\\
0	0.206	2.47151313279425\\
0	0.255	2.44572830121252\\
0	0.304	2.42446688698285\\
0	0.353	2.40782598184273\\
0	0.402	2.39588302757357\\
0	0.451	2.38869440093444\\
0	0.5	2.38629436111989\\
0	0.549	2.38869440093445\\
0	0.598	2.39588302757357\\
0	0.647	2.40782598184274\\
0	0.696	2.42446688698285\\
0	0.745	2.44572830121252\\
0	0.794	2.47151313279425\\
0	0.843	2.50170636385257\\
0	0.892	2.53617702001556\\
0	0.941	2.57478031759191\\
0	0.99	2.61735991847433\\
0.3	0.01	3.30963438692053\\
0.3	0.059	2.95921536607192\\
0.3	0.108	2.67658942626959\\
0.3	0.157	2.62009235376408\\
0.3	0.206	2.57727460330159\\
0.3	0.255	2.5384232602835\\
0.3	0.304	2.50369529574317\\
0.3	0.353	2.47323626631913\\
0.3	0.402	2.44717805499582\\
0.3	0.451	2.42563671532876\\
0.3	0.5	2.40871048893705\\
0.3	0.549	2.39647806475278\\
0.3	0.598	2.3889971433952\\
0.3	0.647	2.38630336110639\\
0.3	0.696	2.38840961529804\\
0.3	0.745	2.39530581859723\\
0.3	0.794	2.40695909129721\\
0.3	0.843	2.4233143844574\\
0.3	0.892	2.44429550876826\\
0.3	0.941	2.46980652885516\\
0.3	0.99	2.49973346992739\\
0.6	0.01	4.09292423041287\\
0.6	0.059	3.72883365058499\\
0.6	0.108	3.37034047823292\\
0.6	0.157	3.01902144482979\\
0.6	0.206	2.74221404265823\\
0.6	0.255	2.67401084067168\\
0.6	0.304	2.62283903643636\\
0.6	0.353	2.57978375968098\\
0.6	0.402	2.54068496033424\\
0.6	0.451	2.50570023373252\\
0.6	0.5	2.47497590097178\\
0.6	0.549	2.44864474554626\\
0.6	0.598	2.42682384923882\\
0.6	0.647	2.40961259700518\\
0.6	0.696	2.39709091952963\\
0.6	0.745	2.38931783724485\\
0.6	0.794	2.3863303609039\\
0.6	0.843	2.38814279160052\\
0.6	0.892	2.39474644811729\\
0.6	0.941	2.40610983257865\\
0.6	0.99	2.42217922773193\\
0.9	0.01	4.90207912164511\\
0.9	0.059	4.52803180337732\\
0.9	0.108	4.15749426881325\\
0.9	0.157	3.79145971650849\\
0.9	0.206	3.43129168295545\\
0.9	0.255	3.07886848847803\\
0.9	0.304	2.80814360443238\\
0.9	0.353	2.73530154344034\\
0.9	0.402	2.67522499879691\\
0.9	0.451	2.6255999274119\\
0.9	0.5	2.58230774946426\\
0.9	0.549	2.54296208524204\\
0.9	0.598	2.50772114576166\\
0.9	0.647	2.47673200807131\\
0.9	0.696	2.45012834804117\\
0.9	0.745	2.42802826818129\\
0.9	0.794	2.41053229017634\\
0.9	0.843	2.3977215809879\\
0.9	0.892	2.38965647672744\\
0.9	0.941	2.38637536002642\\
0.9	0.99	2.38789393463519\\
1.2	0.01	5.73220130317519\\
1.2	0.059	5.35120225854674\\
1.2	0.108	4.97226830927113\\
1.2	0.157	4.59597357976401\\
1.2	0.206	4.22311542800114\\
1.2	0.255	3.85479246580022\\
1.2	0.304	3.49258530912158\\
1.2	0.353	3.13877169107034\\
1.2	0.402	2.87415097232051\\
1.2	0.451	2.79886395060169\\
1.2	0.5	2.73367909744018\\
1.2	0.549	2.67727696897834\\
1.2	0.598	2.62837709876173\\
1.2	0.647	2.5848465352592\\
1.2	0.696	2.54525459992266\\
1.2	0.745	2.5097579995812\\
1.2	0.794	2.47850455871586\\
1.2	0.843	2.45162883740865\\
1.2	0.892	2.42924995135037\\
1.2	0.941	2.41146955228516\\
1.2	0.99	2.39837003790119\\
1.5	0.01	6.57896023693625\\
1.5	0.059	6.19330894478066\\
1.5	0.108	5.8087897104057\\
1.5	0.157	5.42570486354921\\
1.5	0.206	5.04447516697283\\
1.5	0.255	4.6656828983791\\
1.5	0.304	4.29015387414861\\
1.5	0.353	3.91911076572965\\
1.5	0.402	3.55437699438099\\
1.5	0.451	3.19876036247528\\
1.5	0.5	2.94018969856429\\
1.5	0.549	2.8633970108508\\
1.5	0.598	2.79493464618025\\
1.5	0.647	2.73396726988596\\
1.5	0.696	2.67964715451015\\
1.5	0.745	2.63117088283533\\
1.5	0.794	2.5874000795498\\
1.5	0.843	2.54756246913536\\
1.5	0.892	2.51181076275322\\
1.5	0.941	2.48029352378436\\
1.5	0.99	2.45314618832926\\
1.8	0.01	7.43869777362361\\
1.8	0.059	7.05007279306982\\
1.8	0.108	6.66201680332051\\
1.8	0.157	6.27467433630059\\
1.8	0.206	5.88824207566724\\
1.8	0.255	5.50299173777147\\
1.8	0.304	5.11930731479876\\
1.8	0.353	4.73775364935253\\
1.8	0.402	4.35917065913558\\
1.8	0.451	3.98486366117041\\
1.8	0.5	3.61693282116468\\
1.8	0.549	3.25888847361408\\
1.8	0.598	3.00624416751931\\
1.8	0.647	2.92843214936853\\
1.8	0.696	2.85769674753858\\
1.8	0.745	2.79346838722548\\
1.8	0.794	2.73515390119655\\
1.8	0.843	2.68217188167943\\
1.8	0.892	2.63397352011851\\
1.8	0.941	2.58996834469786\\
1.8	0.99	2.54988565748494\\
2.1	0.01	8.30842592222889\\
2.1	0.059	7.91800398626916\\
2.1	0.108	7.52783533870799\\
2.1	0.157	7.1379799117419\\
2.1	0.206	6.74851826019672\\
2.1	0.255	6.35956019382271\\
2.1	0.304	5.97125692560677\\
2.1	0.353	5.58383190992751\\
2.1	0.402	5.19760728976878\\
2.1	0.451	4.81307918048526\\
2.1	0.5	4.43103906636227\\
2.1	0.549	4.05281233790456\\
2.1	0.598	3.68074641711819\\
2.1	0.647	3.31926743506192\\
2.1	0.696	3.07230765518796\\
2.1	0.745	2.99376015250127\\
2.1	0.794	2.92135877995978\\
2.1	0.843	2.85469331011207\\
2.1	0.892	2.79332984537642\\
2.1	0.941	2.7368334895513\\
2.1	0.99	2.68478327091351\\
2.4	0.01	9.18576693729829\\
2.4	0.059	8.79434162426817\\
2.4	0.108	8.40300898471651\\
2.4	0.157	8.01178902027155\\
2.4	0.206	7.62070838853308\\
2.4	0.255	7.22980273884335\\
2.4	0.304	6.83911882151782\\
2.4	0.353	6.44872865731844\\
2.4	0.402	6.05872837412374\\
2.4	0.451	5.66926432969016\\
2.4	0.5	5.28056310495946\\
2.4	0.549	4.89299347166483\\
2.4	0.598	4.50719184412803\\
2.4	0.647	4.12433977290939\\
2.4	0.696	3.74684080160618\\
2.4	0.745	3.38016980580746\\
2.4	0.794	3.13837678944259\\
2.4	0.843	3.05927373101712\\
2.4	0.892	2.98560017113559\\
2.4	0.941	2.91704804079829\\
2.4	0.99	2.85328840712804\\
2.7	0.01	10.0688639024132\\
2.7	0.059	9.67695006920334\\
2.7	0.108	9.28505925963692\\
2.7	0.157	8.89319527230061\\
2.7	0.206	8.50136384708369\\
2.7	0.255	8.10957250888906\\
2.7	0.304	7.71782924511808\\
2.7	0.353	7.32615177178252\\
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Envelope_Logistic_L2_3d.tex
% This file was created by matlab2tikz.
% Minimal pgfplots version: 1.3
%
%The latest updates can be retrieved from
%  http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz
%where you can also make suggestions and rate matlab2tikz.
%
\begin{tikzpicture}

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width=\smallplotwidth,
every outer x axis line/.append style={black},
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1.5	0.696	10.1720196286257\\
1.5	0.745	9.59532126133192\\
1.5	0.794	9.21070774541357\\
1.5	0.843	9.02235241775957\\
1.5	0.892	9.03669477704927\\
1.5	0.941	9.26293960578485\\
1.5	0.99	9.71401700137495\\
1.8	0.01	47.9665594047763\\
1.8	0.059	43.2176037521034\\
1.8	0.108	38.8713261442325\\
1.8	0.157	34.9266777312464\\
1.8	0.206	31.3701724855734\\
1.8	0.255	28.1774290157726\\
1.8	0.304	25.3176838221648\\
1.8	0.353	22.7588242826928\\
1.8	0.402	20.4711289852793\\
1.8	0.451	18.4292624061607\\
1.8	0.5	16.6128968976896\\
1.8	0.549	15.0065221394874\\
1.8	0.598	13.5988883478699\\
1.8	0.647	12.3823606639512\\
1.8	0.696	11.3523303760143\\
1.8	0.745	10.506747600522\\
1.8	0.794	9.84579908109211\\
1.8	0.843	9.37174100548562\\
1.8	0.892	9.08890184923606\\
1.8	0.941	9.00389237734354\\
1.8	0.99	9.1261061049176\\
2.1	0.01	53.6338023609514\\
2.1	0.059	48.5649119601364\\
2.1	0.108	43.8843464184709\\
2.1	0.157	39.597026839571\\
2.1	0.206	35.6978354234844\\
2.1	0.255	32.1705872552828\\
2.1	0.304	28.9903021546726\\
2.1	0.353	26.1274276244033\\
2.1	0.402	23.5520109055228\\
2.1	0.451	21.2366290806745\\
2.1	0.5	19.1579069698519\\
2.1	0.549	17.2969746982209\\
2.1	0.598	15.6393007433301\\
2.1	0.647	14.1742368994699\\
2.1	0.696	12.8944866736046\\
2.1	0.745	11.7956130050099\\
2.1	0.794	10.8756415657911\\
2.1	0.843	10.1347848363138\\
2.1	0.892	9.57530146293878\\
2.1	0.941	9.20151024205835\\
2.1	0.99	9.01999864730153\\
2.4	0.01	59.5442017161251\\
2.4	0.059	54.1731477748457\\
2.4	0.108	49.1760525219774\\
2.4	0.157	44.5611109280935\\
2.4	0.206	40.3294409793808\\
2.4	0.255	36.4725029715402\\
2.4	0.304	32.9721396732038\\
2.4	0.353	29.8032296713166\\
2.4	0.402	26.9375039387737\\
2.4	0.451	24.3469770243491\\
2.4	0.5	22.0062451266001\\
2.4	0.549	19.8936437547382\\
2.4	0.598	17.9915805520528\\
2.4	0.647	16.2863848341417\\
2.4	0.696	14.7679305871045\\
2.4	0.745	13.4291956456712\\
2.4	0.794	12.2658492694031\\
2.4	0.843	11.2759161125698\\
2.4	0.892	10.4595411962883\\
2.4	0.941	9.81887254405496\\
2.4	0.99	9.35808289823134\\
2.7	0.01	65.6826387634306\\
2.7	0.059	60.0242760976006\\
2.7	0.108	54.7266998720892\\
2.7	0.157	49.7993050968507\\
2.7	0.206	45.2471645707357\\
2.7	0.255	41.067943416341\\
2.7	0.304	37.2502953483041\\
2.7	0.353	33.7746920523374\\
2.7	0.402	30.6162046592492\\
2.7	0.451	27.7478699864209\\
2.7	0.5	25.1434960981091\\
2.7	0.549	22.7794661010505\\
2.7	0.598	20.6356214166742\\
2.7	0.647	18.6954993752521\\
2.7	0.696	16.9461945775725\\
2.7	0.745	15.3780421007028\\
2.7	0.794	13.9842488353288\\
2.7	0.843	12.760546498415\\
2.7	0.892	11.7049067720127\\
2.7	0.941	10.8173414397588\\
2.7	0.99	10.0998046627168\\
3	0.01	72.0373754482657\\
3	0.059	66.1037463802307\\
3	0.108	60.5195132949438\\
3	0.157	55.2939076054804\\
3	0.206	50.433989921662\\
3	0.255	45.9418083428683\\
3	0.304	41.8119663630357\\
3	0.353	38.0308763421303\\
3	0.402	34.5781273103323\\
3	0.451	31.4292213288333\\
3	0.5	28.558490070455\\
3	0.549	25.9413786915733\\
3	0.598	23.5558619846517\\
3	0.647	21.3831171952473\\
3	0.696	19.4076885972137\\
3	0.745	17.6173576487608\\
3	0.794	16.0028730255413\\
3	0.843	14.5576393615064\\
3	0.892	13.2774235501209\\
3	0.941	12.1601124543545\\
3	0.99	11.2055426311782\\
};
\addplot3 [solid]
 table[row sep=crcr] {%
-3	0	11.0308481766876\\
-2.7	0	9.97457647018026\\
-2.4	0	9.28667855869521\\
-2.1	0	9.00736520184689\\
-1.8	0	9.17761065516444\\
-1.5	0	9.83523174688171\\
-1.2	0	11.0102219101055\\
-0.9	0	12.7203465611771\\
-0.6	0	14.9682440383992\\
-0.3	0	17.7409767821673\\
0	0	21.0121870002916\\
0.3	0	24.7461352613279\\
0.6	0	28.9024118542134\\
0.9	0	33.4401881325274\\
1.2	0	38.3213331162991\\
1.5	0	43.5122357486808\\
1.8	0	48.9845252206075\\
2.1	0	54.7150234870172\\
2.4	0	60.6852555267979\\
2.7	0	66.8807655349405\\
3	0	73.290398846685\\
};
 \addplot3 [solid]
 table[row sep=crcr] {%
-3	1	73.290398846685\\
-2.7	1	66.8807655349405\\
-2.4	1	60.6852555267978\\
-2.1	1	54.7150234870172\\
-1.8	1	48.9845252206075\\
-1.5	1	43.5122357486808\\
-1.2	1	38.3213331162991\\
-0.9	1	33.4401881325274\\
-0.6	1	28.9024118542134\\
-0.3	1	24.7461352613279\\
0	1	21.0121870002916\\
0.3	1	17.7409767821673\\
0.6	1	14.9682440383992\\
0.9	1	12.7203465611771\\
1.2	1	11.0102219101055\\
1.5	1	9.83523174688171\\
1.8	1	9.17761065516444\\
2.1	1	9.00736520184689\\
2.4	1	9.28667855869521\\
2.7	1	9.97457647018026\\
3	1	11.0308481766876\\
};
 \end{axis}
\end{tikzpicture}%

manuscript.tex
\documentclass[colorlinks=false]{jmlrcia} % Include author names

% The following packages will be automatically loaded:
% amsmath, amssymb, natbib, graphicx, url, algorithm2e

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% 0.32 results in overflow with my latex environement
\newcommand{\smallplotwidth}{0.33\textwidth}


\title[Convexification of Learning from Constraints]{Convexification of Learning from Constraints}
\usepackage{times}
 % Use \Name{Author Name} to specify the name.
 % If the surname contains spaces, enclose the surname
 % in braces, e.g. \Name{John {Smith Jones}} similarly
 % if the name has a "von" part, e.g \Name{Jane {de Winter}}.
 % If the first letter in the forenames is a diacritic
 % enclose the diacritic in braces, e.g. \Name{{\'E}louise Smith}

 % Two authors with the same address
  % \coltauthor{\Name{Author Name1} \Email{abc@sample.com}\and
  %  \Name{Author Name2} \Email{xyz@sample.com}\\
  %  \addr Address}

 % Three or more authors with the same address:
 % \coltauthor{\Name{Author Name1} \Email{an1@sample.com}\\
 %  \Name{Author Name2} \Email{an2@sample.com}\\
 %  \Name{Author Name3} \Email{an3@sample.com}\\
 %  \addr Address}


 % Authors with different addresses:
 \coltauthor{\Name{Iaroslav Shcherbatyi} \Email{shcherbatyi@mpi-inf.mpg.de}\\
 \addr Max Planck Institute for Informatics, Saarbr\"ucken, Germany
% \addr Address 1
 \AND
 \Name{Bjoern Andres} \Email{andres@mpi-inf.mpg.de}\\
 \addr Max Planck Institute for Informatics, Saarbr\"ucken, Germany\\[-5ex] % ??? micro typesetting
 }

\begin{document}

\maketitle

\begin{abstract}
Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning.
%
For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. 
%
In this form, the problem is resistant to standard optimization techniques.
%
We construct MIPs with the same solutions whose objective functions are convex.
%
Specifically, we characterize the tightest convex extension of the objective function, 
given by the Legendre-Fenchel biconjugate.
%
Computing values of this tightest convex extension is NP-hard.
%
However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently.
%
For some decompositions, common loss and regularization functions, we derive a closed form.
\end{abstract}

% \begin{keywords}
% Semi-supervised learning, unsupervised learning, convex extension
% \end{keywords}



\section{Introduction}

We study an optimization problem:
Given 
a finite set $S \neq \emptyset$ whose elements are to be labeled as 0 or 1,
a non-empty set $Y \subseteq \{0,1\}^S$ of \emph{feasible labelings},
$x: S \to \mathbb{R}^m$ with $m \in \mathbb{N}$, called a \emph{feature matrix},
$l: \mathbb{R} \times [0,1] \to \mathbb{R}_0^+$ with 
$l(\cdot, 0)$ convex,
$l(\cdot, 1)$ convex,
$l(r, 1) \rightarrow 0$ as $r \rightarrow \infty$ 
and $l(-r, 0) \rightarrow 0$ as $r \rightarrow \infty$, 
called a \emph{loss function},
$\Theta \subseteq \mathbb{R}^m$ convex, called the set of \emph{feasible parameters},
$\omega: \Theta \to \mathbb{R}_0^+$ convex, called a \emph{regularization function},
and $C \in \mathbb{R}_0^+$, called a \emph{regularization constant},
we consider the optimization problem 
%
\begin{align}
\inf_{(\theta, y) \in \Theta \times Y} 
	\varphi(\theta, y)
\label{eq:problem-main}
\end{align}
%
\vspace{-1ex}
with
\vspace{-1ex}
%
\begin{align}
\varphi(\theta, y)\ :=\ 
    & \omega(\theta) + \frac{C}{|S|} \sum_{s \in S} l_s(\theta, y_s) \\
l_s(\theta, y_s)\ :=\ 
    & l(\langle x_s, \theta \rangle, y_s) 
\enspace .
\end{align}
%
A minimizer $(\hat\theta, \hat y)$, if it exists, defines a \emph{classifier}
$c: \mathbb{R}^m \to \{0,1\}: x \mapsto \frac{1}{2}(1 + \sgn \langle \hat\theta, x \rangle)$ 
and a feasible labeling $\hat y \in Y$.
%
The optimization problem \eqref{eq:problem-main} is a remarkably general abstraction of learning.
On the one hand, it generalizes supervised, semi-supervised and unsupervised learning:
%
\begin{itemize}
\itemsep0ex
\item If the labeling $y$ is fixed by $|Y| = 1$ to precisely one feasible labeling,
\eqref{eq:problem-main} 
specializes to \emph{regularized empirical risk minimization with fixed labels} and is called \emph{supervised}.
%
\item If $Y$ fixes the label of at least one but not all elements of $S$, \eqref{eq:problem-main} is called \emph{semi-supervised}.
%
\item If $Y$ constrains the joint labelings of $S$ without fixing the label of any single element of $S$, \eqref{eq:problem-main} is called \emph{unsupervised}.
For example, consider $Y = \{y \in \{0,1\}^S\ |\ \sum_{s \in S} y_s = \lfloor |S|/2 \rfloor \}$.
\end{itemize}
%
On the other hand, the optimization problem \eqref{eq:problem-main} generalizes classification, clustering and ranking:
%
\begin{itemize}
\itemsep0ex
\item If $S = A \times B$ and $Y$ is the set of characteristic functions of \emph{maps} from $A$ to $B$, \eqref{eq:problem-main} is an abstraction of \emph{multi-label classification}, as discussed, for instance, by 
\citet{joachims-1999,joachims-2003,chapelle-2005-semi,xu-2005-multi-class,chapelle-2006-branch,chapelle-2006-continuation,chapelle-2008-optimization}.
%
\item If $S = A \times A$ and $Y$ is the set of characteristic functions of \emph{equivalence relations} on $A$, \eqref{eq:problem-main} is an abstraction of \emph{clustering}, as discussed by 
\cite{finley-2005,xu-2005-clustering}. 
For fixed parameters $\theta$, 
it specializes to the NP-hard minimum cost clique partition problem
\citep{groetschel-1989,chopra-1993}
that is also known as correlation clustering
\citep{bansal2004correlation,demaine-2006}.
%
\item If $S = A \times A$ and $Y$ is the set of characteristic functions of \emph{linear orders} on $A$, \eqref{eq:problem-main} is an abstraction of \emph{ranking}. 
For fixed parameters $\theta$, 
it specializes to the NP-hard linear ordering problem
\citep{marti-2011}.
\end{itemize}

The set $Y$ of feasible labelings, a subset of the vertices of the unit hypercube $[0,1]^S$, is a non-convex subset of $\mathbb{R}^S$ iff $2 \leq |Y|$.
Typically, one considers a relaxation $y \in P$ of the constraint $y \in Y$ with $P$ a convex polytope such that $\conv\,Y \subseteq P \subseteq [0,1]^S$ and $P \cap \{0,1\}^S = Y$.
In practice, one considers a polytope that is described as an intersection of half-spaces, i.e., by $n \in \mathbb{N}$, $A \in \mathbb{R}^{n \times S}$ and $b \in \mathbb{R}^n$ such that $P = \{y \in \mathbb{R}^S\,|\,A y \leq b \}$.
% Examples are given in Appendix~\ref{section:polytopes}.
Also typically, the set $\Theta \subseteq \mathbb{R}^m$ of feasible parameters is a convex polyhedron and is described also as an intersection of half-spaces, i.e., by $n' \in \mathbb{N}$, $A' \subseteq \mathbb{R}^{n' \times m}$ and $b' \in \mathbb{R}^{n'}$ such that $\Theta = \{\theta \in \mathbb{R}^m\,|\,A' \theta \leq b'\}$.
Hence, the optimization problem 
\eqref{eq:problem-main}
assumes the form of a mixed integer program (MIP):
%
\begin{align}
\inf_{(\theta, y) \in \mathbb{R}^m \times [0,1]^S} \quad
    & \varphi(\theta, y) 
    \label{eq:mip-objective}\\
\textnormal{subject to} \quad
    & A' \theta \leq b' \label{eq:mpi-theta-polytope} \\
    & A y \leq b \label{eq:mip-label-polytope}\\
    & y \in \{0,1\}^S \label{eq:mip-integrality}
\end{align}

\begin{table}[b]
\vspace{-2ex} % ??? micro typesetting
\centering
\small
\caption{Loss functions}
\label{table:loss-functions}
\vspace{1ex}
\begin{tabular}{lll}
\toprule
Loss 
	& Form of $l_s(\theta, y_s)$ 
	& Function $l_s: \mathbb{R}^m \times [0,1] \to \mathbb{R}_0^+$ \\
\midrule
Squared difference 
	& $(\langle \theta, x_s \rangle - y_s)^2$ 
	& convex \\
Logistic 
	& $\log(1 + \exp(-(2 y_s - 1) \langle \theta, x_s \rangle))$ 
	& non-convex \\
Hinge 
	& $\max \{0, 1 - (2 y_s - 1) \langle \theta, x_s \rangle\}$ 
	& non-convex \\	
Squared Hinge 
	& $\max \{0, 1 - (2 y_s - 1) \langle \theta, x_s \rangle\}^2$
	& non-convex \\
\bottomrule
\end{tabular}
\end{table}
%
For \emph{convex} loss functions $l_s$ such as the squared difference loss 
(Tab.~\ref{table:loss-functions}),
the objective function $\varphi$ is convex on the domain $\mathbb{R}^m \times [0,1]^S$.
Thus, the continuous relaxation \eqref{eq:mip-objective}--\eqref{eq:mip-label-polytope}
of the problem \eqref{eq:mip-objective}--\eqref{eq:mip-integrality}
is a convex problem.
Its solutions $(\hat \theta', \hat y')$, although possibly fractional in the coordinates of $y'$, can inform a search for feasible solutions of 
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality}
with certificates
\citep{chapelle-2006-branch,chapelle-2008-optimization,bojanowski2013finding}. 
See also
\citet{bonami2012algorithms}
for a recent survey of convex mixed-integer non-linear programming.
%
For \emph{non-convex} loss functions $l_s$ such as the logistic loss, the Hinge loss and the squared Hinge loss
(Tab.~\ref{table:loss-functions}),
$\varphi$ is non-convex on the domain $\mathbb{R}^m \times [0,1]^S$.
In this case, 
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality}
is resistant to standard optimization techniques.
% In particular, targeted experiments by
% \citet{chapelle-2006,chapelle2008optimization}
% have shown that feasible solutions of \eqref{eq:problem-main} found by local search can define a poor classifier while the optimal solution of \eqref{eq:problem-main} defines a classifier that is flawless.
See
\citet{tawarmalani-2004,lee-2011,belotti-2013}
for an overview of non-convex mixed-integer non-linear programming.



\subsection{Contribution}
%
\begin{figure*}
\subfigure[]{
\label{fig:mip-non-convex}
\input{figures/ext_ncvx.tex}
}\hfill
\subfigure[]{
\label{fig:mip-trivial}
\input{figures/ext_triv.tex}
}\hfill
\subfigure[]{
\label{fig:mip-convex-envelope}
\input{figures/ext_cvx.tex}
}
\caption{Depicted above in (a) is the non-convex objective function $\varphi$ 
of the optimization problem
\eqref{eq:problem-main}
for $Y = \{0,1\}$, $\Theta = \mathbb{R}$,
the Hinge loss 
(Tab.~\ref{table:loss-functions})
and 
$\omega(\cdot) = \|\cdot\|_2^2$. 
Its restriction $\phi$ to the feasible set $\Theta \times \{0,1\}$ is depicted in black.
As the goal is to minimize $\varphi$ over the feasible set,
one can replace the values of $\varphi$ for $y \in (0,1)$ without affecting the 
solution, for instance, by zero, as depicted in (b).
In this paper, we characterize the tightest convex extension of $\phi$ to $\Theta \times \conv Y$,
which is depicted in (c).}
\label{fig:ext_description}
\end{figure*} 
%
We construct,
for the MIP
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality}
whose objective function is non-convex,
MIPs with the same solutions whose objective functions are convex.
Our approach is illustrated in 
Fig.~\ref{fig:ext_description}
and is summarized below.

In Section~\ref{section:relaxation},
we consider the restriction $\phi$ of $\varphi$ to the feasible set $\Theta \times Y$
and characterize the tightest convex extension $\phi^{**}$ of $\phi$ to $\Theta \times \conv\,Y$.
This tightest convex extension $\phi^{**}$ is mostly of theoretical interest as computing its values is NP-hard. 
% However, our characterization is constructive and defines not just the tightest convex extension but a class of convex extensions, each with a different trade-off between tightness and computational complexity. 

In Section~\ref{section:characterization}, 
we consider a decomposition of the function $\phi$ into a sum of functions.
By applying our characterization of tightest convex extensions to every function in this sum,
we construct a convex extension $\phi'$ of $\phi$ to $\Theta \times [0,1]^S$ which is not generally tight but whose values can be computed efficiently.
For common loss and regularization functions, we derive a closed form.

For every convex extension $\phi'$ we construct, including $\phi^{**}$,
the MIP
%
\begin{align}
\inf_{(\theta, y) \in \mathbb{R}^m \times \mathbb{R}^S} \quad
    & \phi'(\theta, y) 
    \label{eq:convex-mip-objective}\\
\textnormal{subject to} \quad
    & A' \theta \leq b' \label{eq:convex-mpi-theta-polytope} \\
    & A y \leq b \label{eq:convex-mip-label-polytope}\\
    & y \in \{0,1\}^S \label{eq:convex-mip-integrality}
\end{align}
%
has the same solutions as
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality}.
Like 
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality},
it is NP-hard, due to the integrality constraint
\eqref{eq:convex-mip-integrality}.
Unlike
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality},
its objective function and polyhedral relaxation 
\eqref{eq:convex-mip-objective}--\eqref{eq:convex-mip-label-polytope}
are convex. 
Thus, unlike
\eqref{eq:mip-objective}--\eqref{eq:mip-integrality},
it is accessible to a wide range of standard optimization techniques.



\section{Related Work}

\subsection{Convex Extensions}
%
For large classes of univariate and bivariate functions, 
tightest convex extensions have been characterized by
\citet{tawarmalani2013explicit}
and
\citet{locatelli2014technique}.
Convex envelopes of multivariate functions that are convex in all but one variable have been characterized by
\citet{jach2008convex}.
For functions of the form $f(x,y) = g(x) h(y)$ and with $g$ and $h$ having additional properties,
e.g., $g$ being component-wise concave and submodular, and $h$ being univariate convex,
tightest convex extensions have been characterized by
\cite{khajavirad2012convex, khajavirad2013convex}.
%
For functions $f: A \times B \to \mathbb{R}$ with $A, B \subseteq \mathbb{R}^n$ 
and $f(a,\cdot)$ being either convex or concave
and $f(\cdot,b)$ being either convex or concave
for any $a \in A$ and $b \in B$,\
tightest convex extensions have been characterized by
\citet{ballerstein2013convex}.
%
Tightest convex extensions of pseudo-Boolean functions $f: \{0,1\}^n \to \mathbb{R}$
are known as convex closures
\citep{MAL-039}.
Convex closures of submodular functions are Lov\'asz extensions.

The characterization of tightest convex extensions 
of functions $f: \Theta \times Y \to \mathbb{R}$ with $f(\cdot, y)$ convex for all $y \in Y$
that we establish is consistent with results of
\citet{jach2008convex}
for functions $f: \mathbb{R}^n \times \{0,1\} \to \mathbb{R}$.
It extends some results of
\citet{jach2008convex}
to non-differentiable $f(\cdot, 0)$ and $f(\cdot, 1)$. 



\subsection{Regularized Empirical Risk Minimization with Constrained Labels}
%
Regularized empirical risk minimization with constrained labels
has been studied intensively in the special case of semi-supervised learning for 01-classification.
%
Algorithms that find feasible solutions efficiently are due to
\citet{joachims-1999,joachims-2003,chapelle-2005-semi,chapelle-2006-continuation,chapelle-2006-branch,sindhwani2006deterministic}.
%
A branch-and-bound algorithm that solves the problem to optimality was suggested by
\citet{vapnik1974theory}
and has been implemented and applied to data by
\citet{chapelle-2006-branch,chapelle-2008-optimization},
with the result that optimal solutions generalize better typically than feasible solutions found by approximate algorithms.

An approach to the problem by convex optimization, specifically, by a semi-definite relaxation of the dual problem, was proposed by
\citet{debie-2006}.
Similar relaxations have been studied in the context of semi-supervised learning for multi-label classification
\citep{xu-2005-multi-class,guo2011adaptive},
correlation clustering
\citep{xu-2005-clustering,zhang2009maximum}
and latent variable estimation
\citep{guo2008convex}.
For maximum margin clustering, a convex relaxation tighter than the SDP relaxation is constructed by \citet{li-2009}.
%
For multi-label classification with a softmax loss function, a tight SDP relaxation is proposed by \cite{joulin2012convex}.




\newpage
\section{Tightest Convex Extensions}
\label{section:relaxation}
%
In this section, we consider the restriction $\phi$ of $\varphi$ to the feasible set $\Theta \times Y$,
i.e., the function 
%
\begin{align}
\phi: 
    \quad \Theta \times Y \to \mathbb{R}_0^+: 
    \quad (\theta,y) \mapsto \varphi(\theta, y)
\enspace .
\end{align}
%
We characterize the tightest convex extension $\phi^{**}$ of $\phi$ to $\Theta \times \conv\,Y$
in Theorem~\ref{theorem:main}.
%
\begin{definition}
\citep{tawarmalani2002convexification}
For any $n \in \mathbb{N}$, any $A \subseteq \mathbb{R}^n$ and any $\phi: A \to \mathbb{R}$, 
a function $\phi': \conv\,A \to \mathbb{R}$ is called a \emph{convex extension} of $\phi$ 
iff $\phi'$ is convex and $\forall a \in A: \phi'(a) = \phi(a)$.
% 
Moreover, a function $\phi^{**}: \conv\,A \to \mathbb{R}$ is called the \emph{tightest convex extension} of $\phi$ 
iff $\phi^{**}$ is a convex extension of $\phi$ and, 
for every convex extension $\phi'$ of $\phi$ 
and for all $a \in \conv\,A$: $\phi'(a) \leq \phi^{**}(a)$.
\end{definition}

\begin{lemma}
A (tightest) convex extension $\phi^{**}$ of $\phi$ exists.
\end{lemma}

\begin{theorem} 
\label{theorem:main} 
For every finite $S \neq \emptyset$,
every $Y \subseteq \{0,1\}^S$ with $|Y| > |S|$,
every $m \in \mathbb{N}$, 
every convex $\Theta \subseteq \mathbb{R}^m$
and every $\phi: \Theta \times Y \to \mathbb{R}_0^+$ such that $\phi(\cdot, y)$ is convex for every $y \in Y$,
%
the tightest convex extension
$\phi^{**}: \Theta \times \conv\,Y \to \mathbb{R}_0^+$ of $\phi$ 
is such that for all $(\theta,y) \in \Theta \times \conv\,Y$:
%
\begin{align}
\phi^{**}(\theta, y)
    = \min_{Y' \in \mathcal{Y}(y)} 
        \inf_{\theta': Y' \to \Theta} \Bigg\{ 
            \sum_{y' \in Y'} \lambda_{y Y'}(y') \, \phi(\theta'(y'), y')
            \ \Bigg|
            \sum_{y' \in Y'} \lambda_{y Y'}(y') \, \theta'(y') = \theta
        \Bigg\}
\label{eq:main-theorem-1}
\end{align}
%
where
%
\begin{align}
\mathcal{Y}(y) := \left\{
    Y' \in \tbinom{Y}{|S| + 1}
    \ \middle|\  
    y \in \conv\,Y'
\right\}
\end{align}
%
is the set of $(|S| + 1)$-elementary subsets $Y'$ of $Y$ having $y$ in their convex hull,
and, for every $y \in \conv\,Y$ and every $Y' \in \mathcal{Y}(y)$: 
$\lambda_{y Y'}: Y' \to \mathbb{R}_0^+$ are the coefficients in the convex combination of $y$ in $Y'$, i.e.
%
\begin{align}
\sum_{y' \in Y'} \lambda_{y Y'}(y') \, y' & = y \\
\sum_{y' \in Y'} \lambda_{y Y'}(y') & = 1
\enspace .
\end{align}
\end{theorem}
%
Remarks are in order:
%
\begin{itemize}
\itemsep0ex
\item $|\mathcal{Y}(y)|$ can be strongly exponential in $|S|$.
This is expected, given that the complexity of computing the value of a convex extension is NP-hard. 
%
\item Given a polynomial-time oracle for the outer minimization in
\eqref{eq:main-theorem-1},
the tightest convex extension can be computed in polynomial time.
%
\item Conceivable is a polynomial-time oracle for constraining $\mathcal{Y}(y)$ to a strict subset without compromising optimality in 
\eqref{eq:main-theorem-1}.
%
\item An upper bound on the tightest convex extension can be obtained by considering any subset of $\mathcal{Y}(y)$.
\end{itemize}



\section{Efficient Convex Extensions} 
\label{section:characterization}
%
\begin{figure*}
\subfigure[]{
\label{fig:L1-nonconvex}
\input{figures/Envelope_H1_L1_ORG_3d.tex}
}\hfill
\subfigure[]{
\label{fig:L1-discontinious}
\input{figures/Envelope_H1_L1_3d.tex}
}\hfill
\subfigure[]{
\label{fig:L1-bounded-continuous}
\input{figures/Envelope_H1_L1_BND_3d.tex}
}
\caption{Depicted above in (a) is the non-convex objective function $\varphi$ 
of the optimization problem
\eqref{eq:problem-main}
for $Y = \{0,1\}$, $\Theta = \mathbb{R}$, the Hinge loss 
(Tab.~\ref{table:loss-functions})
and $\omega(\cdot) = \|\cdot\|_1$.  
Its restriction $\phi$ to $\Theta \times \{0,1\}$ is depicted in black.
Depicted in (b) is the tightest convex extension of $\phi$ to $\Theta \times [0,1]$.
Depicted in (c) is the tightest convex extension of $\phi$ to $[-b,b] \times [0,1]$, for $b \in \mathbb{R}_0^+$.
It can be seen that, for bounded $\theta$, the tightest convex extension is continuous.}
\label{fig:L1-explanation}
\end{figure*} 
%
In this section, we characterize, for common loss and regularization functions, convex extensions $\phi'$ of $\phi$ that are not necessarily tight but can be computed efficiently in general.
The construction is in two steps:

Firstly, we consider an additive decomposition of $\phi$.
Specifically, we consider a convex function $c: \Theta \to \mathbb{R}_0^+$ and, for every $s \in S$, a function $d_s: \Theta \times \{0,1\} \to \mathbb{R}_0^+$ such that $d_s(\cdot, 0)$ and $d_s(\cdot, 1)$ are convex.
These functions are chosen such that, for all $(\theta, y) \in \Theta \times Y$:
%
\begin{align}
\phi(\theta, y) = c(\theta) + \frac{1}{|S|} \sum_{s \in S} d_s(\theta, y_s)
\enspace .
\label{eq:decomposition}
\end{align}
%
One example is given by $c := \omega$ and, for every $s \in S$, $d_s := C \, l_s$.
Another example is given by $c := 0$ and, for every $s \in S$, every $\theta \in \Theta$ and every $y_s \in \{0,1\}$: $d_s(\theta, y_s) := \omega(\theta) + C l_s(\theta, y_s)$.

Secondly, we characterize, for each $d_s$, its tightest convex extension $d^{**}_s: \Theta \times [0,1] \to \mathbb{R}$, by a specialization of 
Theorem~\ref{theorem:main}
stated as
Corollary~\ref{theorem:binary_envelope_theorem}.
A convex extension $\phi'$ of $\phi$ to $\Theta \times [0,1]^S$ is then given by 
%
\begin{align}
\phi': 
    \quad \Theta \times [0,1]^S \to \mathbb{R}_0^+:
    \quad (\theta, y) \mapsto c(\theta) + \frac{1}{|S|} \sum_{s \in S} d^{**}_s(\theta, y_s)
\enspace .
\end{align}

\begin{corollary} 
\label{theorem:binary_envelope_theorem} 
For every $d: \Theta \times \{0,1\} \to \mathbb{R}_0^+$ 
such that $d_0(\cdot) := d(\cdot, 0)$ and $d_1(\cdot) := d(\cdot, 1)$ are convex, 
the tightest convex extension $d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+$ of $d$ is such that 
for all $(\theta, y) \in \Theta \times Y$:
%
\begin{align}
\label{eq:binary_convex_envelope}
d^{**}(\theta, y) = \begin{cases} 
d(\theta, y) &\mbox{if } y \in \{0,1\} \\ 
\Psi(\theta, y) & \mbox{if } y \in (0,1)
\end{cases}
\end{align}
%
with
%
\begin{align}
\Psi(\theta, y) 
= 
\inf_{\theta^0, \theta^1 \in \Theta} \left\{
    (1-y) d_0(\theta^0)+y d_1(\theta^1)
    \ \middle| \ 
    (1-y) \theta^0 + y \theta^1 = \theta
\right\}
\enspace .
\label{eq:big_psi}
\end{align}
\end{corollary}

The solutions of the optimization problem \eqref{eq:big_psi} are characterized in Lemma~\ref{lemma:subgradient-1}.
Subgradients are given in 
Lemma~\ref{corollary:subgradient}.
%
\begin{lemma} 
\label{lemma:subgradient-1}
For every solution $(\hat\theta^0, \hat\theta^1) \in \Theta^2$ of \eqref{eq:big_psi}, 
there exists a $G \neq \emptyset$ such that, for $\xi_{\theta, y}: \Theta \to \Theta: t \mapsto (\theta-(1-y) t)/y$:
%
\begin{align}
(\delta d_0)(\hat\theta^0) \ \bigcap\ \ (\delta d_1)(\xi_{\theta, y}(\hat\theta^0)) & = G
    \label{eq:minimizer_convex_envelope}\\
(\delta d_0)(\xi_{\theta, 1-y}(\hat\theta^1)) \ \bigcap\ \ (\delta d_1)(\hat\theta^1) & = G
    \label{eq:minimizer_convex_envelope_equivalent}
\enspace .
\end{align}
\end{lemma}
%
\begin{lemma}
\label{corollary:subgradient}
If a solution $(\hat\theta^0, \hat\theta^1) \in \Theta^2$ of \eqref{eq:big_psi} exists,
then 
$\tbinom{v}{w} \in (\delta d^{**}) (\theta, y)$
iff
$v \in G$ with $G$ defined equivalently in 
\eqref{eq:minimizer_convex_envelope} and \eqref{eq:minimizer_convex_envelope_equivalent}
and $w = v^T\hat\theta^0 - v^T \hat\theta^1 + d_1(\hat\theta^1) - d_0(\hat\theta^0)$.
Otherwise, $y \in \{0,1\}$ and:
\begin{align}
y = 0 
    \quad \Rightarrow \quad
    \forall v \in (\delta d(\cdot, 0))(\theta): \ 
        & \tbinom{v}{\infty} \in (\delta d^{**}) (\theta, y) \\
y = 1
    \quad \Rightarrow \quad
    \forall v \in (\delta d(\cdot, 1))(\theta): \ 
        & \tbinom{v}{-\infty} \in (\delta d^{**}) (\theta, y)
\enspace .
\end{align}
\end{lemma}

We proceed as follows:
In Section~\ref{section:simple-examples},
we consider two decompositions of the class
\eqref{eq:decomposition}
for which the tightest convex extension of any $d_s$ is easy to characterize but the resulting convex extension of $\phi$ is rather loose.
In Sections~\ref{section:l2} and \ref{section:l1},
we consider decompositions of the class
\eqref{eq:decomposition}
with $c = 0$.
These yield the tightest convex extension of $\phi$ of any decomposition of the class
\eqref{eq:decomposition}.
For any combination of the logistic loss, the Hinge loss and the squadred Hinge loss 
(Tab.~\ref{table:loss-functions})
with either L1 regularization $\omega(\cdot) = \|\cdot\|_1$ or L2 regularization  $\omega(\cdot) = \|\cdot\|_2^2$, we characterize the convex extension explicitly and, in some cases, in closed form.
Examples of all combinations are depicted in 
Fig.~\ref{fig:all_envelopes}.



\subsection{Instructive Examples}
\label{section:simple-examples}

The first convex extension we characterize is for the decomposition 
\eqref{eq:decomposition} 
with $c := \omega$ and $\forall s \in S: d_s := C l_s$.
% It is used, e.g., by \citet{chapelle-2008-optimization}.

\begin{corollary}
\label{corollary:trivial-convex-envelope}
For any $d: \Theta \times \{0,1\} \to \mathbb{R}_0^+$ 
such that $d_0(\cdot) := d(\cdot, 0)$ and $d_1(\cdot) := d(\cdot, 1)$ are convex
and for which $d_0(-r) \to 0$ and $d_1(r) \to 0$ as $r \to \infty$,
the convex extension $d^{**}: \Theta \times [0,1] \to \mathbb{R}$ of $d$ has the form 
%
\begin{align}
d'(\theta, y) = \begin{cases} 
d(\theta, y) &\mbox{if } y \in \{0,1\} \\ 
0 & \mbox{if } y \in (0,1)
\end{cases} 
\enspace .
\end{align}
\end{corollary}

The second convex extension we characterize is for the logistic loss
$l_s(\theta, y) = - \langle x, \theta \rangle y + \log(1+e^{\langle x,\theta \rangle})$,
for $\omega(\cdot) = \|\cdot\|_2^2$
and for the decomposition 
\eqref{eq:decomposition} 
such that, for every $s \in S$,
$d_s(\theta, y) := \|\theta\|_2^2 - C \langle x, \theta \rangle y$.

\begin{corollary}
\label{corollary:instructive-2}
The tightest convex extension 
$d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+$
of
$d: \Theta \times \{0,1\} \to \mathbb{R}_0^+$
with
$d(\theta, y) = \| \theta \|_2^2 - C \langle x, \theta \rangle y$
has the form
$d^{**}(\theta, y) = ||\theta -y \frac{C}{2} x ||_2^2 -y \|\frac{C}{2} x\|_2^2$.
\end{corollary}



\subsection{L2 Regularization}
\label{section:l2}
%
We now consider $\omega(\cdot) = \frac{1}{2} \|\cdot\|_2^2$
and decompositions
\eqref{eq:decomposition}
such that, for every $s \in S$: $d_s(\theta, y) = \omega(\theta) + C\,l_s(\theta, y)$.

\begin{corollary} 
\label{corollary:L2-general}
The tightest convex extension $d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+ $ of $ d : \Theta \times \{0,1\} \to \mathbb{R}_0^+ $ with $d(\theta,y) = \tfrac{1}{2} ||\theta||_2^2 + C l ( \langle x , \theta \rangle , y )$ is given by 
\eqref{eq:binary_convex_envelope} and \eqref{eq:big_psi}.
Moreover, the solution of \eqref{eq:big_psi} is given 
by $\hat\theta^0 = \theta - y C x z$ with
%
\begin{align}
z \  \in \ 
(\delta l_0)(x^T(\theta-y C x z))
- (\delta l_1)(x^T(\theta+(1-y) C x z)) 
\  \subseteq \  \mathbb{R}
\enspace .
\label{eq:L2-general}
\end{align}
\end{corollary}

Although this characterization is not a closed form, values of $d^{**}$ can be computed efficiently using the bisection method. For specific loss functions, closed forms are derived below.
%
\begin{corollary}
\label{corollary:H1L2}
The tightest convex extension
$d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+ $ 
of 
$d: \Theta \times \{0,1\} \to \mathbb{R}_0^+ $ 
with $d(\theta,y) = \tfrac{1}{2} ||\theta||_2^2 + C l ( \langle x , \theta \rangle , y )$
and $l : \mathbb{R} \times \{0,1\} \to \mathbb{R}_0^+$ of the form
%
\begin{align}
l(r, y) = (C_0(1-y) + C_1 y) \max \{0, 1 - (2y-1) r\}
\end{align}
%
where $C_0, C_1 \in \mathbb{R}_+$ define weights on the two values of $y$,
is given by 
Corollary~\ref{corollary:L2-general}.
Moreover, every $z$ satisfying
\eqref{eq:L2-general} 
holds
%
\begin{align}
z \in \left\{
0, \ 
C_0, \ 
C_1, \ 
C_0 + C_1, \ 
\frac{1+x^T \theta}{ y C x^T x}, \ 
\frac{1-x^T \theta}{(1-y) C x^T x}
\right\}
\enspace .
\end{align}
\end{corollary}
%
\begin{corollary}
\label{corollary:H2L2}
The tightest convex extension
$d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+ $ 
of 
$d : \Theta \times \{0,1\} \to \mathbb{R}_0^+$ 
with
%
$d(\theta, y) = \tfrac{1}{2} ||\theta||_2^2 + C l ( \langle x , \theta \rangle , y )$
%
and $l : \mathbb{R} \times \{0,1\} \to \mathbb{R}_0^+$ of the form
%
\begin{align}
l(r, y) = \frac{C_0(1-y) + C_1 y}{2} (\max \{0, 1 - (2y-1) r\})^2
\end{align}
%
where $C_0, C_1 \in \mathbb{R}_+$ define weights on the two values of $y$,
is given by 
Corollary~\ref{corollary:L2-general}.
Moreover, every $z$ satisfying
\eqref{eq:L2-general} 
holds
%
\begin{align}
z \in \left\{
0, \ 
\frac{C_0 + C_0 x^T \theta}{1 +  C_0 y C x^T x}, \ 
\frac{C_1-C_1 x^T \theta }{ 1+  C_1 (1-y) C x^T x}, \ 
\frac{C_0+C_1+(C_0-C_1) x^T \theta}{1+(C_0  y + C_1 (1-y)) C x^T x}
\right\}
\enspace .
\end{align}
\end{corollary}



\subsection{L1 regularization}
\label{section:l1}
%
We now consider $\omega(\cdot) = \|\cdot\|_1$
and decompositions
\eqref{eq:decomposition}
such that, for every $s \in S$:
$d_s(\theta, y) = \omega(\theta) + C\,l_s(\theta, y)$. 
We focus on a special case where $\theta$ is bounded. 
This is necessary in order for the convex extension to be continuous;
see Fig.~\ref{fig:L1-explanation}.

\begin{corollary} 
\label{corollary:L1-general}
For $b, t \in (\mathbb{R} \cup \{-\infty, \infty\})^m$ 
and $\Theta = \{\theta \in \mathbb{R}^m : b \leq \theta \leq t \}$,
the tightest convex extension $d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+$ 
of $d: \Theta \times \{0,1\} \to \mathbb{R}_0^+$ 
with $d(\theta,y) = ||\theta||_1 + C l (\langle x , \theta \rangle , y)$ 
%
is given by \eqref{eq:binary_convex_envelope} and \eqref{eq:big_psi}.
Moreover, the solution of \eqref{eq:big_psi} is given by
%
\begin{align}
\hat\theta^0 = \begin{cases}
    \theta' 
        & \mbox{if } \exists r \in  a(x^T \theta'): ||r C x ||_\infty \leq 2 \\
    \textnormal{aux}(\theta', x, a, b, t)
        & \mbox{otherwise}
\end{cases}
\end{align}
%
with $a: \mathbb{R} \to 2^\mathbb{R}: p \mapsto (\delta l_0)( p) - (\delta l_1)( (x^T \theta-(1-y) p)/y)$
and
%
\begin{align}
\label{eq:L1-projection-th}
\forall i \in \{1,\ldots,m\}: 
\quad
\theta'_i = \begin{cases} 
    \min\limits_{r \in [b'_i, t'_i]} |r| &\mbox{if } \theta_i > 0 \Leftrightarrow x_i > 0\\ 
    \min\limits_{r \in [b'_i, t'_i]} \left| \dfrac{\theta_i}{(1-y)} - r \right| &\mbox{otherwise}
\end{cases}
\enspace .
\end{align}
\end{corollary}
%
The function ``aux'' is defined below in terms of Alg.~\ref{algorithm:aux}.
%
At its core, this algorithms solves, for fixed $k \in [m]$ and fixed $V \in \Theta$,
the equation \eqref{eq:L1-general} for the unknown $z \in \mathbb{R}$.
For specific loss functions, closed forms are derived 
in Corollaries~\ref{corollary:H1L1} and \ref{corollary:H2L1}.
%
\begin{align}
\left| a\left(x^T V - z x_k\right) C x_k \right| = 2 
\label{eq:L1-general}
\end{align}
%
\begin{corollary}
\label{corollary:H1L1}
The tightest convex extension $d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+$ 
of $d: \Theta \times \{0,1\} \to \mathbb{R}_0^+$ 
with $d(\theta,y) = ||\theta||_1 + C l (\langle x , \theta \rangle , y)$
and $l: \mathbb{R} \times \{0,1\} \to \mathbb{R}_0^+$ of the form
%
\begin{align}
l(r, y) = (C_0(1-y) + C_1 y) \max \{0, 1 - (2y-1) r \}
\end{align}
%
where $C_0, C_1 \in \mathbb{R}_+$ define weights on the two values of $y$,
is given by Corollary~\ref{corollary:L1-general}.
Moreover, every solution $z \in \mathbb{R}$ of \eqref{eq:L1-general} holds
%
\begin{align}
z \in \left\{
\frac{1+x^T V}{x_k^2}, \ 
\frac{y+x^T (\theta - (1-y) V)}{(1-y) x_k^2}
\right \}
\enspace .
\end{align}
\end{corollary}
%
\begin{corollary}
\label{corollary:H2L1}
The tightest convex extension $d^{**}: \Theta \times [0,1] \to \mathbb{R}_0^+$ 
of $d: \Theta \times \{0,1\} \to \mathbb{R}_0^+$ 
with $d(\theta,y) = ||\theta||_1 + C l (\langle x, \theta \rangle , y)$
and $l : \mathbb{R} \times \{0,1\} \to \mathbb{R}_0^+$ of the form
%
\begin{align}
l(r, y) = \frac{C_0(1-y) + C_1 y}{2} (\max \{0, 1 - (2y-1) r \})^2
\end{align}
%
where $C_0, C_1 \in \mathbb{R}_+$ define weights on the two values of $y$,
is given by Corollary~\ref{corollary:L1-general}.
Moreover, every solution $z \in \mathbb{R}$ of \eqref{eq:L1-general} holds
\eqref{eq:z} with $v_0 := x^T V$ and $v_1 := - x^T \xi_{\theta,y}(V)$.
%
\begin{align}
z \in \bigg\{ &
	\frac{1+v_0 - 2 (C |x_k| C_0)^{-1}}{x_k^2}, \  
	\frac{1+v_1 - 2 (C |x_k| C_1)^{-1}}{x_k^2}, \nonumber\\
	&
	\frac{ y(C_0 (1 + v_0) + C_1 (1+v_1) - 2 (C |x_k|)^{-1}) }{(y C_0 + (1-y) C_1) x_k^2}
	\bigg\}
\label{eq:z}
\end{align}
\end{corollary}

\begin{algorithm}
\KwIn{$ \theta',x \in \mathbb{R}^n, a: \mathbb{R} \to 2^\mathbb{R}, C \in \mathbb{R}_0^+, b,t \in (\mathbb{R} \cup \{-\infty, \infty\})^n $}
\KwOut{$V \in \Theta$ }
$V := \theta' $ \;

$\forall i \in [n]: b'_i := \max\{b_i, (\xi_{\theta, 1-y}(t))_i\}$ \;

$\forall i \in [n]: t'_i := \min\{t_i, (\xi_{\theta, 1-y}(b))_i\}$ \;

$I \in \mathbb{N}^n $ such that $ |x_{I_1}| \geq |x_{I_2}| \geq ... \geq |x_{I_n}|  $ \;

\For{$j = 1 ... n $}{
    \If{$ \exists r \in a(x^T V): || r C x ||_\infty \leq 2 $}{
   		\Return $ V $\;
    } 
    $ i := I_j $\;

    \eIf{$ x_i > 0 $}{
        $ V_i := b'_i $\;
    }{
        $ V_i := t'_i $\;
    }
    \If{$ \exists r \in a(x^T V): || r C x ||_\infty \leq 2 $}{
        $V_i := V_i - z x_k $ with $ z \in \mathbb{R} $ such that $|a(x^T V - z x_k ) C x_k| = 2$\;

        \Return $ V $\;
    }
}
\Return $ V $\;
\caption{Computation of the function ``aux''}
\label{algorithm:aux}
\end{algorithm}



\section{Conclusion}
\label{section:conclusion}
%
We have characterized convex extensions, including the tightest convex extension,
of functions $f: \Theta \times Y \to \mathbb{R}_0^+$
with $\Theta \subseteq \mathbb{R}^m$ convex,
$Y \subseteq \{0,1\}^n$
and $f(\cdot, y)$ convex for every $y \in \{0,1\}^n$.
This has allowed us to state regularized empirical risk minimization with constrained labels as a mixed integer program whose objective function is convex.
Convex extensions that strike a practical balance between tightness and computational complexity are a topic of future work.

\begin{figure*}
\subfigure[]{%
\label{fig:Envelope_Logistic_L2-plot}
\input{figures-2/Envelope_Logistic_L2_3d.tex}
}\hfill
\subfigure[]{%
\label{fig:Envelope_Logistic_L1-plot}
\input{figures-2/Envelope_Logistic_L1_3d.tex}
}\hfill
\subfigure[]{%
\label{fig:Envelope_Logistic_L1_bounded-plot}
\input{figures-2/Envelope_Logistic_L1_BND_3d.tex}
}\\
\subfigure[]{%
\label{fig:Envelope_H1_L2-plot}
\input{figures-2/Envelope_H1_L2_3d.tex}
}\hfill
\subfigure[]{%
\label{fig:Envelope_H1_L1-plot}
\input{figures-2/Envelope_H1_L1_3d.tex}
}\hfill
\subfigure[]{%
\label{fig:Envelope_H1_L1_bounded-plot}
\input{figures-2/Envelope_H1_L1_BND_3d.tex}
}\\
\subfigure[]{%
\label{fig:Envelope_H2_L2-plot}
\input{figures-2/Envelope_H2_L2_3d.tex}
}\hfill
\subfigure[]{%
\label{fig:Envelope_H2_L1-plot}
\input{figures-2/Envelope_H2_L1_3d.tex}
}\hfill
\subfigure[]{%
\label{fig:Envelope_H2_L1_bounded-plot}
\input{figures-2/Envelope_H2_L1_BND_3d.tex}
}
\caption{Depicted above are tightest convex extensions of 
the logistic loss (first row), 
the Hinge loss (second row) and 
the squared Hinge loss (third row),
%
in conjunction with 
%
L2 regularization (first column),
L1 regularization (second column) and
L1 regularization with $\theta$ constrained to $[-3.1, 3.1]$ (third column).
%
Values for $y \in \{0,1\}$ are depicted in black.
%
Note that the convex extensions of L1-regularized loss functions for unbounded $\theta$ are discontinuous. 
%
Parameters for these examples are $x = 1$ and
\ref{fig:Envelope_Logistic_L2-plot}: $ C = 16 $, 
\ref{fig:Envelope_Logistic_L1-plot}: $ C = 5 $,
\ref{fig:Envelope_Logistic_L1_bounded-plot}: $ C = 5 $,
%
\ref{fig:Envelope_H1_L2-plot}: $ C = 16 $, 
\ref{fig:Envelope_H1_L1-plot}: $ C = 5 $, 
\ref{fig:Envelope_H1_L1_bounded-plot}: $ C = 5 $,
%
\ref{fig:Envelope_H2_L2-plot}: $ C = 4 $, 
\ref{fig:Envelope_H2_L1-plot}: $ C = 4 $.
\ref{fig:Envelope_H2_L1_bounded-plot}: $ C = 4 $.}
\label{fig:all_envelopes}
\end{figure*} 






% Acknowledgments---Will not appear in anonymized version
% \acks{We thank a bunch of people.}



\bibliography{manuscript}



\end{document}



\appendix
\section{Proofs}
\label{section:proofs}



\subsection{Proof of Theorem~\ref{theorem:main}}

Let $(\theta, y) \in \Theta \times \conv Y$, 
let $(Y', \theta')$ be a solution of \eqref{eq:main-theorem-1} with objective value $\psi(\theta, y)$
and let $\phi'$ be the tightest convex extension of $\phi$ to $\Theta \times \conv Y$.

Firstly, we have $\phi'(\theta, y) \leq \Psi(\theta, y)$ as
\begin{align}
\Psi(\theta, y) = \sum_{y' \in Y'} \alpha_{y Y'}(y') \, \phi(\theta'(y'), y')
\end{align}
is a convex combination of values of $\phi$.

In order to establish $\phi'(\theta, y) \geq \Psi(\theta, y)$,
it is sufficient to show that every supporting affine function $f: \Theta \times \conv Y \to \mathbb{R}$ of $\phi$ that is tight at $(\theta, y)$ holds $f(\theta, y) \geq \Psi(\theta, y)$.

This, in turn, follows from the fact that \eqref{eq:main-theorem-1} is an optimization over the values of all non-redundant supporting affine functions of $\phi$ at $(\theta, y)$.


The $|S| + 1$ points $\{((\theta'(y'), y'), \phi(\theta'(y'), y')\,|\,y' \in Y'\}$ defined by the solution $(Y', \theta')$ of \eqref{eq:main-theorem-1} define the \emph{class} of affine functions containing these points.



Let $n := |S|$.
For every $(\theta, y) \in \Theta \times Y$
and $(Y', \theta')$ a solution to 
\eqref{eq:main-theorem-1}

\begin{align}
\label{eq:convex_envelope_combination}
\phi^{**}(\theta, y) = \sum_{ y' \in Y'} \alpha_{yY'}(y') \phi(\theta'(y'), y')
\end{align}
%
which is a convex combination of 
$ n + 1 $ 
values of $ \phi $ computed for $ n + 1 $ affinely independent labelings
$ Y' \in Y^{n+1} $ and values of $ \theta'(y') \in \Theta $,
indexed by the $ y' $. Affinity of a supporting affine function implies also that
%
\begin{align}
y &= \sum_{ y' \in Y'} \alpha_{yY'}(y') y',\\
\theta &= \sum_{ y' \in Y'} \alpha_{yY'}(y') \theta'(y')
\end{align}
%
Notice that given only values $ Y' \in Y^{n+1} $, 
values 
$ \alpha_{yY'} \in [0,1]^{n+1} $ 
can be computed by solving the system of linear equations
%
\begin{align}
\label{eq:convex_envelope_linear}
y &= \sum_{ y' \in Y'} \alpha_{yY'}(y') y',\\
1 &= \sum_{ i \in 0 ... n} \alpha_{yY'}(y')
\end{align}
%
We consider only 
$ \alpha_{yY'}(y')\in [0,1] $ 
as any other values of $ \alpha_{yY'}(y') $ 
imply that 
$ y $ 
is outside of the convex hull of 
$ Y' $.
We are looking for a facet of the lower convex hull of $\phi$,
whose projection on the domain of $\phi$ contains $y$.
Every such facet is defined by some values $ Y' \in Y^{n+1} $, 
and respective values $\theta \in \Theta$.
Thus we simply try all $ Y' \in Y^{n+1} $ which
can define such facet and which contain $ y $ in their convex hull.
Notice that any combination of points in the convex hull is an upper bound on 
the lower convex hull, thus minimizing $ Y' \in Y^{n+1} $
defines facet of the lower convex hull which defines value 
of the convex extension.
We define the set of all such feasible $ Y' \in Y^{n+1} $ as
%
\begin{align}
\mathcal{Y}(y) := \{ Y' \in \dbinom{Y}{|S|+1} | y \in conv(Y') \}
\end{align}
%

Above imply that in order to compute 
the value of convex extension for some point 
$ \{\theta, y \} \in \Theta \times conv(Y) $
one can consider all combinations of 
$ \{\theta', Y' \} \in \Theta^{n+1} \times \mathcal{Y}(y)  $
such that for values of 
$ \alpha_{yY'} $ 
computed from the system \eqref{eq:convex_envelope_linear} it holds
%
\begin{align}
y &= \sum_{ y' \in Y'} \alpha_{yY'}(y') y',\\
\theta &= \sum_{ y' \in Y'} \alpha_{yY'}(y') \theta'(y') ,\\
\bigcap_{y' \in Y'} &(\delta_\theta \phi) (\theta'(y'), y') \neq 0
\end{align}
and among all of them select one 
which yields the smallest value of 
\eqref{eq:convex_envelope_combination}.

Notice that any convex combination of some values 
$ \{\theta', Y' \} \in \Theta^{n+1} \times \mathcal{Y}(y)  $
for which above requirements hold yield 
an upper bound on the value of the convex extension.

Now we show that for some given 
$ Y' \in \mathcal{Y}(y) $ ,
values of 
$ \theta' \in \Theta^{n+1} $
that yield minimal value of \eqref{eq:convex_envelope_combination}
can be found by solving the convex optimization problem of the form
%
\begin{align}
\inf\limits_{\theta':Y' \to \Theta} &\ \sum_{ y' \in Y'} \alpha_{yY'}(y') \phi(\theta'(y'), y') \\ 
	 & \mbox{Subject to}  \\
	 & \theta = \sum_{ y' \in Y'} \alpha_{yY'}(y') \theta'(y')
\end{align}
%
where values of 
$ \alpha_{yY'} $ 
are defined by \eqref{eq:convex_envelope_linear}.

The only requirement that is not satisfied explicitly 
by solution of above optimization problem is that 
%
\begin{align}
\label{eq:convex_envelope_requirement}
\bigcap_{y' \in Y'} &(\delta_\theta \phi) (\theta'(y'), y') \neq 0.
\end{align}
%
However, consider the dual of above optimization problem:
%
\begin{align}
\max\limits_{\beta \in \mathbb{R}^{n}} \inf\limits_{\theta':Y' \to \Theta} \ \sum_{y' \in Y'} \alpha_{y Y'}(y') [\phi(\theta'(y'), y') + \beta^T \theta'(y') ] - \beta^T \theta 
\end{align}
%
and as the primal optimization problem contains independent variables 
$ \theta^i $
this simplifies to
%
\begin{align}
\max\limits_{\beta \in \mathbb{R}^{n}} \ \sum_{y' \in Y'} \inf\limits_{\theta'(y') \in \Theta} \alpha_{yY'}(y') [\phi(\theta'(y'), y') + \beta^T \theta'(y') ] - \beta^T \theta \\ 
\end{align}
%
and thus for every solution $ \theta'(y') $ it holds that 
%
\begin{align}
(\delta_\theta \phi)(\theta'(y'), y') = -\beta
\end{align}
%
which shows that for solutions of stated convex optimization requirement 
\eqref{eq:convex_envelope_requirement} 
also holds. This concludes the proof.



\subsection{Proof of Corollary~\ref{theorem:binary_envelope_theorem}}

Corollary~\ref{theorem:binary_envelope_theorem} 
is a special case of the Theorem~\ref{theorem:main}.
To see that, notice that for a function with a single 
binary argument there is only one combination of
values $  Y' \in \mathcal{Y}(y) $, where $ \mathcal{Y}(y) $ is 
defined in the Theorem~\ref{theorem:main}, and that 
value of $ y \in [0,1] $ can be expressed as a convex combination
$
y = y 1 + (1-y) 0
$.
Thus a characterization of the convex extension given by 
the Theorem~\ref{theorem:main}
requires solving a single convex optimization problem
where values of a convex combination
are $ (y, 1-y) $.



\subsection{Proof of Corollary~\ref{corollary:trivial-convex-envelope}}
%
Notice that 
$ \forall  (\theta, y) \in \mathbb{R}^n \times (0,1) $ 
it holds that 
%
\begin{align}
v^T \xi_{\theta, y}(-a v) \geq  v^T( \frac{\theta + (1-y) a v}{y} ) \geq 0 \\
 v^T(-a v) \leq 0
\end{align}
%
for sufficiently large 
$ a > 0 $ 
and 
$ \xi_{\theta, y} $ 
defined in the
Corollary \ref{theorem:binary_envelope_theorem}. 
Using the sum law for limits, it holds 
$ \forall  (\theta, y) \in \mathbb{R}^n \times (0,1)  $
%
\begin{align}
\lim\limits_{a \to \infty} (1-y)l_0( -a v) + y l_1(\xi_{\theta, y}(-a v) ) \to 0 \enspace .
\end{align}
%
Notice, that 
the function in the limit operator is exactly 
$ \psi_{\theta, y}(-a v) $ 
defined in 
the Corollary \ref{theorem:binary_envelope_theorem}, 
and as 
$ l > 0 $ 
and 
$ v \in \Theta $
%
\begin{align}
\inf\limits_{\theta_0 \in \Theta} \psi_{\theta, y}(\theta_0) = \Psi(\theta, y) = 0 \enspace .
\end{align}
%


\subsection{Proof of Corollary~\ref{corollary:instructive-2}} 
%
Subgradients are 
\begin{align}
(\delta d_0)(\theta) &= 2\theta \\ 
(\delta d_1)(\theta) &= 2\theta -2 x'
\end{align}
%
In order to derive a closed form of the convex extension $d'$ of $d$, 
one needs to derive the minimizer $\theta^{**}$ of 
\eqref{eq:big_psi} 
in the Corollary \ref{theorem:binary_envelope_theorem}. 
%
Substituting the subgradients of $d_0$ and $d_1$ in the optimality condition 
\eqref{eq:minimizer_convex_envelope}
yields
%
\begin{align}
2 \theta^{**} = 2 \left( \frac{\theta-(1-y)\theta^{**}}{y} \right) -2 x'
\end{align}
%
and so
%
\begin{align}
\theta^{**} = \theta - y x' \enspace .
\end{align}
%
Substituting $\theta^{**}$ in 
\eqref{eq:big_psi} 
yields, for the function $\Psi$ in 
the Corollary~\ref{theorem:binary_envelope_theorem}:
%
\begin{align}
\Psi(\theta, y) 
&= y d_1( \xi_{\theta, y}( \theta - y x' ))+(1-y) d_0(\theta - y x') \\
&= ||\theta -y x' ||_2^2 - y \|x'\|_2^2
\end{align}
%
Note that values of $\Psi$ coincide with $d$ for $y \in \{0,1\}$. 
Thus, $d' = \Psi$.



\subsection{Proof of Corollary~\ref{corollary:L2-general}}
%
With the case distinction subgradient of $ d_0(\cdot) = d(\cdot, 0) $ and  $ d_1(\cdot) = d(\cdot, 1) $ can be expressed as
%
\begin{align}
(\delta d_0) (\theta) &= \theta + C x g_0(\theta)\\
(\delta d_1) (\theta) &= \theta + C x g_1(\theta )
\end{align}
%
%
Using the optimality condition \eqref{eq:minimizer_convex_envelope} and by multiplying by $ y $ we get
%
\begin{align}
\theta^{**} + y C x g_0(\theta^{**}) = \theta + y C x g_1(\xi_{\theta, y}(\theta^{**}) )
\end{align}
%
\begin{align}
\label{d-l2:subgradient-equation}
\theta^{**} = \theta - y C x( g_0(\theta^{**}) - g_1(\xi_{\theta, y}(\theta^{**}) )) \enspace .
\end{align}
%
Notice that above equation implies that $ \theta^{**} = \theta - y C x z $, where $ z \in \mathbb{R}, V = - C x $. It holds that
%
\begin{align}
\xi_{\theta, y}(\theta + y V z) &=  \theta - (1-y) V z \enspace .
\end{align}
%
We define 
%
\begin{align}
s_0(z) &= g_0( y V z + \theta) \\ 
s_1(z) &= g_1( \theta - (1-y) V z)
\end{align}
%
and then equation \eqref{d-l2:subgradient-equation} turns into
%
\begin{align}
y V z + \theta = \theta + y V( s_0(z)+s_1(z))
\end{align}
%
which is equivalent to 
%
\begin{align}
z = s_0(z)+s_1(z) \enspace .
\end{align}




\subsection{Proof of Corollary~\ref{corollary:H1L2}}
%
For the given loss function $ \Lambda $, we define
%
\begin{align}
g_0(t) &= C_0 \sigma(1+t,0) ,\\
g_1(t) &= -C_1 \sigma(1-t,0) ,\\
\sigma(t) &= \begin{cases} 
0 &\mbox{if } t < 0 \\ 
[0,1] & \mbox{if } t = 0 ,\\
1 & \mbox{if } t > 0
\end{cases}
\end{align}
%
and based on above definitions we define 
%
\begin{align}
s_0(z) &= g_0(y x^T V z + x^T \theta) ,\\
s_1(z) &= -g_1( x^T \theta-(1-y) x^T V z)
\end{align}
%
and thus equation that needs to be solved is 
%
\begin{align}
s_0(z) + s_1(z) - z = 0
\end{align}
%
Notice, that both $ s_0, s_1 $ are piecewise constant, for some $ z $ being zero, and for some being linear. We consider all possible cases of $ s_0, s_1 $ and for each possibility, we find the solution. So, for both $ s_0, s_1 $ being zero, the solution is $ z = 0 $. For the case when $ s_0 $ is not zero, an equation turns into
%
\begin{align}
C_0 + C_0 (y x^T V z_1+ x^T \theta) - z_1 = 0 
\end{align}
%
which yields a possible solution $ z_1 $.

For the case when $ s_1 $ is not zero, stated equation turns into
%
\begin{align}
C_1 + C_1 ((1-y) x^T V z_2- x^T \theta) - z_2 = 0 
\end{align}
%
which yields a possible solution $ z_2 $.

Similarly, we get an expression for $z_3$ for the case when  $s_1, s_0$ are assumed to be not zero.


\subsection{Proof of Corollary~\ref{corollary:H2L2}}
%
For the given loss function $ \Lambda $, we define
%
\begin{align}
g_0(t) &= C_0 \max(1+t,0) ,\\
g_1(t) &= -C_1 \max(1-t,0)
\end{align}
%
and based on above definitions we define 
%
\begin{align}
s_0(z) &= g_0(y x^T V z + x^T \theta) ,\\
s_1(t) &= -g_1( x^T \theta-(1-y) x^T V z)
\end{align}
%
and thus equation that needs to be solved is 
%
\begin{align}
s_0(z) + s_1(z) - z = 0
\end{align}
%
Notice, that both $ s_0, s_1 $ are constant everywhere except for points where the argument of $ \sigma $ in expressions for $ s_0, s_1 $ turns to a zero, where it is multivalued. Thus, we first check if $ s_0(z) + s_1(z) $ intersects with $ z $ at a point where both $ s_0, s_1 $ are constant. As for the points where $ s_0(z) + s_1(z) $ is a constant its range is a finite set $ \{ 0, C1, C2, C1+C2 \} $  this defines a set of values of $ z $ that potentially satisfy $ s_0(z) + s_1(z) = z $. Otherwise, solution is the one of at most two points for which either $ s_0 $ or $ s_1 $ are multivalued. These points are found by solving equations 
%
\begin{align}
1+y x^T V z_1 + x^T \theta &= 0 ,\\
1- (x^T \theta-(1-y) x^T V z_2) \theta &= 0
\end{align} 
%
which define two other solutions.




\subsection{Proof of Corollary~\ref{corollary:L1-general}}
%

%
Here and below a lower index value $i \in [n]$ denotes an index of a value in a vector (except for the functions $ d_0, d_1 $), for example for $v^\theta$ we mean by $ v_i^\theta \in \mathbb{R} $ the value of $ i $ - th element of vector $v^\theta$, which corresponds to the subgradient of L1 norm.
%
With the case distinction the subgradient of $ d_0(\cdot) = d(\cdot, 0) $ and  $ d_1(\cdot) = d(\cdot, 1) $ can be expressed as
%
\begin{align}
(\delta d_0) (\theta) &= g_0(\theta) C x + v^\theta + N_\theta\\
(\delta d_1) (\theta) &= g_1(\theta) C x + v^\theta + N_\theta
\end{align}
%
where
%
\begin{align}
g_0(\cdot) &= (\delta \Lambda_0 )(x^T \cdot) ,\\
g_1(\cdot) &= (\delta \Lambda_1 )(x^T \cdot) ,
\end{align}
%
\begin{align}
v_i^\theta = (\delta || \cdot ||_1^1)(\theta) =  \begin{cases} 
\{-1\} &\mbox{if } \theta_i < 0 \\ 
[-1,1] & \mbox{if } \theta_i = 0 \\
\{1\} & \mbox{if } \theta_i > 0
\end{cases}  ,\quad
 i  \in [n]
\end{align}
%
and $ N $ is a normal cone of a set $ \Theta $.

Using the optimality condition \eqref{eq:minimizer_convex_envelope} and by multiplying by $ y $ we get the optimality condition for $ \theta^{**} $ that allows to compute the value of the convex extension for given $ \theta, y$:
%
\begin{align}
\label{eq:L1-full-eq}
a (\theta^{**}) C x \bigcap v^{\xi_{\theta, y} (\theta^{**})} - v^{\theta^{**}} + N^{\xi_{\theta, y} (\theta^{**})} - N^{\theta^{**}} \neq \emptyset
\end{align}
%
where $ a : \mathbb{R}^n \to \mathbb{R}^\mathbb{R} $ is such that
%
\begin{align}
a(\cdot) = g_0(\cdot)-g_1( \xi_{\theta, y} (\cdot) ) 
\end{align}
%
%
First, consider
%
\begin{align}
z C x \in v^{\xi_{\theta, y} (\theta^{**})} - v^{\theta^{**}}
\end{align}
%
Above inclusion have a solution only when $ z \in \mathbb{R} $ is such that $ ||z C x||_\infty \leq 2$. This is due to the fact that the range of the right hand side is $ [-2,2] $. By Lemma~\ref{lemma:general-eCx-solution} a solution to above equation (when $ || z C x ||_\infty \leq 2 $) is
%
\begin{align}
\theta''_i = \begin{cases} 
0 &\mbox{if } \theta_i > 0 \Leftrightarrow x_i > 0\\ 
\dfrac{\theta_i}{(1-y)} &\mbox{else }
\end{cases} 
\end{align}
%
Above solution is constructed such that either $\xi_{\theta, y} (\theta'')_i$ or $\theta''_i$ turns to zero. 
So for some index $ i \in [n] $ is when $ x_i \geq 0 $ then it holds that
\begin{align}
1 \in v^{\xi_{\theta, y} (\theta'')}_i ,\\
[-1,1] \in -v^{\theta''}_i ,
\end{align}
%
and thus for any 
$ x_i \in [0,2] $ 
it holds that 
$ x_i \in 1 + [-1,1] $. 
Similarly, above solution for some index 
$ i \in [n] $ when 
$ x_i < 0 $ is such that 
%
\begin{align}
[-1,1] \in v^{\xi_{\theta, y} (\theta'')}_i ,\\
-1 \in -v^{\theta''}_i ,
\end{align}
%
and thus for any $ x_i \in [-2,0] $ it holds that $ x_i \in [-1,1] - 1$.  
For a more detailed proof, refer to the Lemma~\ref{lemma:general-eCx-solution}.

Now consider an inclusion of the form
%
\begin{align}
\label{eq:th1-no-ncone-eq}
a (\theta^{**}) C x \bigcap v^{\xi_{\theta, y} (\theta^{**})} - v^{\theta^{**}} \neq \emptyset
\end{align}
%
If $ \exists z \in a (\theta'') $ such that $ ||z C x||_\infty \leq 2 $, then $ \theta'' $ is a solution to above inclusion. Otherwise, let $ k \in \mathbb{N} $ be an index of value in the vector $ x $ such that 
%
\begin{align}
|x_k| = \max\limits_{i \in [n]} |x_i|
\end{align} 
%
To solve the system of inclusions \eqref{eq:th1-no-ncone-eq}, we exploit the following: let $ \theta^{z,k} \in \Theta $ such that
%
\begin{align}
\theta^{z,k}_i = \begin{cases}
\theta''_i - z x_k & \mbox{if } i = k \\
\theta''_i & \mbox{else }
\end{cases}
\end{align}
%
which is a copy of the vector $ \theta'' $ where the element at $ k $ - th index of this vector, which corresponds to the maximum value of the vector $ x $, was decreased by the value of $ z x_k $. 

The function $ a(\theta^{z,k}) $ is decreasing for increasing $ z \in \mathbb{R}_+ $. This follows from the fact that $ x^T \theta^{z,k} = x^T \theta'' - z x_k^2 $ and as $ g_0, -g_1 $ in the definition of $a$ are increasing. 
Also it can be checked that
%
\begin{align}
2 \in v^{\xi_{\theta, y} (\theta^{z,k})}_k - v^{\theta^{z,k}}_k ,\quad \forall z \geq 0, x_k \geq 0
\end{align}
%
and
%
\begin{align}
-2 \in v^{\xi_{\theta, y} (\theta^{z,k})}_k - v^{\theta^{z,k}}_k ,\quad \forall z \geq 0, x_k < 0 \enspace ,
\end{align}
%
that is, increasing $ z $ preserves signs of $ \xi_{\theta, y} (\theta^{z,k}) $ and $ \theta^{z,k} $. This is due to the fact that $ \theta''_k $ and $ x_k $ have different signs for $ z > 0 $.

Notice that for both cases $ |x_k| > 2 $, as it is assumed that \eqref{eq:th1-no-ncone-eq} cannot be satisfied by $ \theta'' $. 
By increasing the value of $ z \in \mathbb{R}_+ $ the value of function $ a(\theta^{z,k}) $ always decreases, and so the value of $ z $ can be found that $ a(\theta^{z,k}) x_k \in v^{\xi_{\theta, y} (\theta^{z,k})}_k - v^{\theta^{z,k}}_k $. Moreover, as $ x_k $ has the maximum absolute value in the vector $ x $, it follows that $ \forall i \in [n], i \neq k $ it holds that $ |C x_i| \leq 2 $, and as only the value of $ k $ - th element in the vector $ \theta'' $ was changed, $i$ - th equation is satisfied by the solution $ \theta^{z,k} $. This implies that  $ \theta^{z,k} $ is a solution to equation \eqref{eq:th1-no-ncone-eq}, where $z \geq 0$ is selected such that
%
\begin{align}
a(\theta^{z,k}) C x_k \bigcap v^{\xi_{\theta, y} (\theta^{z,k})}_k - v^{\theta^{z,k}}_k \neq \emptyset
\end{align} 

Inclusion \eqref{eq:th1-no-ncone-eq} is obtained from the inclusion \eqref{eq:L1-full-eq} by dropping the normal cone function, which corresponds to $ \Theta = \mathbb{R}^n $. For a rectangular subset $ \Theta \subset \mathbb{R}^n $, we do the following: first, a projection of $ \theta'' $ on $ \Theta $, given by the expression for $ \theta' $ \eqref{eq:L1-projection-th}, is computed. It can be checked that it holds $ \forall i \in [n] $
%
\begin{align}
[0,2] \subset v^{\xi_{\theta, y} (\theta')}_i - v^{\theta'}_i + N^{\xi_{\theta, y} (\theta')}_i- N^{\theta'}_i, x_i \geq 0
\end{align}
%
and
%
\begin{align}
[-2,0] \subset v^{\xi_{\theta, y} (\theta')}_i - v^{\theta'}_i + N^{\xi_{\theta, y} (\theta')}_i- N^{\theta'}_i, x_i < 0
\end{align}
%
If the value of $\theta'_i $ is not on the boundary of the feasible region, this holds because the normal cone value is zero and the expression of subgradient is similar to the unbounded case. Otherwise it can be checked that the value of $ v^{\xi_{\theta, y} (\theta')}_i - v^{\theta'}_i $ is zero and the normal cone function is set valued.

Thus if it holds that $ ||a(\theta') C x ||_\infty \leq 2 $ then $ \theta' $ is a solution to \eqref{eq:L1-full-eq}. 

For the case $ ||a(\theta') C x ||_\infty > 2 $ we first define all $ \theta^{**} $ such that the right hand side of \eqref{eq:L1-full-eq} is not empty.  Let
%
\begin{align}
b' = \max(b, \xi_{\theta, 1-y}(t)), \\
t' = \min(t, \xi_{\theta, 1-y}(b)),
\end{align}
%
where the maximum and minimum operation is component-wise. Notice that $ \theta^{**} $ is feasible if and only if $ \forall i \in [n] $ it holds that 
%
\begin{align}
\theta^{**}_i \in [b'_i, t'_i] \enspace . 
\end{align}
%
as otherwise $ \xi_{\theta, y}(\theta^{**}) $ or $ \theta^{**}$ is outside of the box constraint.

Now we show that the 
Algorithm~\ref{algorithm:aux} 
that defines an output of the function ``aux''
allows to obtain a solution to \eqref{eq:L1-full-eq} from $ \theta' $ when $ \forall r \in a(\theta') , || r C x ||_\infty \geq 2 $. Notice that similarly to the unbounded case, it holds that decreasing the value of $ \theta'_i, i \in [n] $ by the value of $ z x_i, z \in \mathbb{R}_+ $ only decreases the value of $ a(\theta) $. Notice that it still holds that
%
\begin{align}
2 \in v^{\xi_{\theta, y} (\theta^{z,k})}_i - v^{\theta^{z,k}}_i ,\quad \forall z \geq 0, x_k \geq 0
\end{align}
%
and
%
\begin{align}
-2 \in v^{\xi_{\theta, y} (\theta^{z,k})}_i - v^{\theta^{z,k}}_i ,\quad \forall z \geq 0, x_k < 0 ,
\end{align}
%
for any $ k \in [n] $. Denote the subgradient value at $ \theta^{**} \in \Theta $ as
%
\begin{align}
\Gamma^{\theta^{**}} = v^{\xi_{\theta, y} (\theta^{**})} - v^{\theta^{**}} + N^{\xi_{\theta, y} (\theta^{**})} - N^{\theta^{**}}
\end{align}
%
Notice that it necessary holds that 
%
\begin{align}
[2, \infty) \in \Gamma^{b'}_i
\end{align}
%
and
%
\begin{align}
(-\infty,2] \in \Gamma^{t'}_i
\end{align}
%
as values of the subgradient for the unbounded case are in $ [-2,2] $, and set values of normal cone are open intervals.

Based on above observations we propose the following approach to obtain a solution from $ \theta' $: first, select $ k $ to be the index of the largest element. Then, decrease the value of $ \theta' $ by $ z x_k $ until either $ \exists r \in a(\theta') , | r C x | = 2 $ or until $ \theta'_k $ is on the boundary. 
If $ \exists r \in a(\theta') , | r C x_k | = 2 $ , then $ \theta' $ is a solution, as for any other element it holds that $ | r C x_k | \leq 2 $. 
Otherwise, if $\theta'_k$ is on the boundary, set $ k $ to the index $ k' $ of the next largest element  and repeat the procedure. 
Notice that an inclusion for $ k $ - th index still holds, because if $ \theta' $ is found such that $ \exists r \in a(\theta') , | r C x_{k'} | = 2 $, it  holds that $ | r C x_k | \geq 2 $ and thus belongs to the values of $ \Gamma^{\theta'}_k $ on the boundary. 
More formally this is formulated in the Algorithm~\ref{algorithm:aux}.
This concludes the proof.
 
\begin{lemma} \label{lemma:general-eCx-solution} Let $ z, C \in \mathbb{R}_+, y \in (0,1), \theta, x \in \Theta $ such that 
%
\begin{align}
\label{eq:eCx-requirement}
|| z C x ||_\infty \leq 2
\end{align}
%
then $ \theta' \in \Theta $ that satisfy 
%
\begin{align}
\label{eq:eCx-vector-equation}
z C x \in v^{\xi_{\theta,y}(\theta')}-v^{\theta'}
\end{align}
%
can be expressed as
%
\begin{align}
\label{eq:eCx-general-solution}
\theta'_i = \begin{cases} 
0 &\mbox{if } pos(\theta_i) = pos(x_i)\\ 
\dfrac{\theta_i}{(1-y)} &\mbox{else }
\end{cases} 
\end{align}
%
where
%
\begin{align}
pos(v) = \begin{cases} 
1 &\mbox{if } v > 0\\ 
-1 &\mbox{else }
\end{cases} 
\end{align}
%
and values of $ v^{\xi_{\theta,y}(\theta')} $, $v^{\theta'} $ that satisfy \eqref{eq:eCx-vector-equation}, for $ i \in [n]$, when $ pos(x_i) = pos(\theta_i) $ can be selected as
%
\begin{align}
pos(x_i) - z C x_i &\in v_i^{\theta'_i} \\
pos(x_i) & \in v_i^{\xi_{\theta,y}(\theta')}
\end{align}
%
and otherwise when $ pos(x_i) \neq pos(\theta_i) $
%
\begin{align}
z C x_i + pos(x_i)  &\in v_i^{\xi_{\theta,y}(\theta')} \\
-pos(x_i) &\in v_i^{\theta'}
\end{align}
%

\end{lemma}
\begin{proof} 
Solution $ \theta' \in \Theta $ to \eqref{eq:eCx-vector-equation} can be determined for each index $ i \in [n] $ independently. 
%
Assume $ pos(x_i) = pos(\theta_i) $. As per provided solution we set $\theta'_i = 0$. For such $ \theta'_i $ it holds that 
%
\begin{align}
v_i^{\xi_{\theta,y}(\theta')} &= v_i^{\frac{\theta_i-(1-y)\theta_i'}{y}} = v_i^{\frac{\theta_i}{y}}\\
pos(x_i) &= pos(\theta_i) \in v_i^{\frac{\theta_i}{y}}  \\
-v_i^{\theta'_i} &= -v_i^{0} \in [-1,1] \enspace .
\end{align}
%
Notice that for such $ \theta'_i $ it holds 
%
\begin{align}
pos(x_i) + [-1,1] \subseteq v_i^{\xi_{\theta,y}(\theta')}-v_i^{\theta'}  \enspace .
\end{align}
%
Also, it holds that
%
\begin{align}
z C x_i \in pos(x_i) + [-1,1]
\end{align}
%
due to the requirement \eqref{eq:eCx-requirement}, thus $ \theta_i' = 0 $ is indeed a solution to \eqref{eq:eCx-vector-equation} when $ pos(x_i) = pos(\theta_i) $. Specifically, one can select
%
\begin{align}
pos(x_i) - z C x_i &\in v_i^{\theta'_i} \\
pos(x_i) & \in v_i^{\xi_{\theta,y}(\theta')}
\end{align}
%
for which it holds that $ pos(x_i)-(pos(x_i) - z C x_i) = z C x_i $.





Assume $ pos(x_i) \neq pos(\theta_i) $. Let $ \theta'_i = \dfrac{\theta_i}{(1-y)} $. For such $ \theta'_i $, it holds that 
%
\begin{align}
v_i^{\xi_{\theta,y}(\theta')} &=  v_i^{\frac{\theta_i-(1-y)\theta'_i}{y}} = v_i^{0} \in [-1,1]\\
-pos(x_i) &\in v_i^{\theta'_i} \enspace .
\end{align}
%
Similarly to the previous case it holds that
%
\begin{align}
[-1,1] + pos(x_i) \subseteq v_i^{\xi_{\theta,y}(\theta')}-v_i^{\theta'} 
\end{align}
%
and thus
%
\begin{align}
z C x_i \in [-1,1] + pos(x_i)
\end{align}
%
therefore $ \theta_i' = \dfrac{\theta_i}{(1-y)} $ is indeed a solution to \eqref{eq:eCx-vector-equation} when $ pos(x_i) \neq pos(\theta_i) $. Specifically, one can select
%
\begin{align}
z C x_i + pos(x_i)  &\in v^{\xi_{\theta,y}(\theta')} \\
-pos(x_i) &\in v_i^{\theta'_i}
\end{align}
%
for which we get that $ (z C x_i + pos(x_i))-pos(x_i) = z C x_i $.

\end{proof}




\subsection{Proof of Corollary~\ref{corollary:H1L1}}
%
For the given function loss function $ \Lambda $, we define
%
\begin{align}
g_0(t) &= C_0 \sigma(1+t,0) ,\\
g_1(t) &= -C_1 \sigma(1-t,0) ,\\
\sigma(t) &= \begin{cases} 
0 &\mbox{if } t < 0 \\ 
[0,1] & \mbox{if } t = 0 ,\\
1 & \mbox{if } t > 0
\end{cases}
\end{align}
%
and based on above definitions we define 
%
\begin{align}
s_0(z) &= g_0(y x^T V z + x^T \theta) ,\\
s_1(t) &= -g_1( x^T \theta-(1-y) x^T V z)
\end{align}
%
and thus an equation that needs to be solved is 
%
\begin{align}
s_0(z) + s_1(z) - z = 0
\end{align}
%
Notice, that both $ s_0, s_1 $ are piecewise constant, for some $ z $ being zero, and for some being linear. We consider all possible cases of $ s_0, s_1 $ and for each possibility, we find a solution. So, for both $ s_0, s_1 $ being zero, the solution is $ z = 0 $. For the case when $ s_0 $ is not zero, equation turns into
%
\begin{align}
C_0 + C_0 (y x^T V z_1+ x^T \theta) - z_1 = 0 
\end{align}
%
which yields a possible solution $ z_1 $.

For the case when $ s_1 $ is not zero, stated equation turns into
%
\begin{align}
C_1 + C_1 ((1-y) x^T V z_2- x^T \theta) - z_2 = 0 
\end{align}
%
which yields a possible solution $ z_2 $.

Similarly, we get an expression for $z_3$ for the case when  $s_1, s_0$ are assumed to be not zero.

\subsection{Proof of Corollary~\ref{corollary:H2L1}}
%
For the given function loss function $ \Lambda $, we define
%
\begin{align}
g_0(t) &= C_0 \max(1+t,0) ,\\
g_1(t) &= -C_1 \max(1-t,0)
\end{align}
%
and based on above definitions we define 
%
\begin{align}
s_0(z) &= g_0(y x^T V z + x^T \theta) ,\\
s_1(t) &= -g_1( x^T \theta-(1-y) x^T V z)
\end{align}
%
and thus an equation that needs to be solved is 
%
\begin{align}
s_0(z) + s_1(z) - z = 0
\end{align}
%
Notice, that both $ s_0, s_1 $ are constant everywhere except for the points where the argument of $ \sigma $ in expressions for $ s_0, s_1 $ turns to zero, where it is multivalued. Thus, we first check if $ s_0(z) + s_1(z) $ intersects with $ z $ at a point where both $ s_0, s_1 $ are constant. As for the points where $ s_0(z) + s_1(z) $ is a constant its range is a finite set $ \{ 0, C1, C2, C1+C2 \} $  this defines a set of values of $ z $ that potentially satisfy $ s_0(z) + s_1(z) = z $. Otherwise, a solution is the one of at most two points for which either $ s_0 $ or $ s_1 $ are multivalued. These points are found by solving equations 
%
\begin{align}
1+y x^T V z_1 + x^T \theta &= 0 ,\\
1- (x^T \theta-(1-y) x^T V z_2) \theta &= 0
\end{align} 
%
which define two other possible solutions.




\subsection{Proof of Lemma~\ref{corollary:subgradient}}
%
For the convex extension characterized by the Theorem \ref{theorem:main} for a special case of the Corollary~\ref{theorem:binary_envelope_theorem} an affine function $ h_{\theta,y}$ 
%
\begin{align}
h_{\theta, y}(t, y') &= (1-y')h_0(t)  + y' h_1(t) ,\\
h_0(t) &= w^T (t - \theta^0) + d_0(\theta^0) ,\\
h_1(t) &= w^T (t - \theta^1) + d_1(\theta^1)
\end{align}
%
%
which can be written as
%
\begin{align}
\label{eq:convex-envelope-subgradient}
h_{\theta, y}(t, y') &= d_0(\theta^{**}_0) - w^T\theta^{**}_0 + w^Tt + g_y y ,\\
g_y &=  w^T\theta^{**}_0 - w^T \theta^{**}_1 + d_1(\theta^{**}_1) - d_0(\theta^{**}_0)
\end{align}
%
is a supporting affine hyperplane. By the definition of supporting affine function, slope of \eqref{eq:convex-envelope-subgradient} is exactly a subgradient of the convex extension.

Notice that for $ y = 1 $ the optimality condition \eqref{eq:minimizer_convex_envelope} turns into
%
\begin{align}
\label{eq:subgradient_1}
( \delta d_0)(\theta^{**}) \bigcap ( \delta d_1)(\theta) = G' \neq \emptyset
\end{align}
%
Solution to the above problem for any of the functions whose convex extensions expressions we derived can be obtained by setting $ y = 1 $ in the expression for the solution of \eqref{eq:minimizer_convex_envelope}, and a support function \eqref{eq:convex-envelope-subgradient} can also be constructed using any $ w \in G' $. Notice however, that a subgradient of the convex function is Minkowski sum of subgradient and normal cone of its domain, thus we obtain that $ g_y = g_y + a$, where depending on which boundary $ y $ lies $ a \in \mathbb{R}_+ $ for $ y = 1 $ and $ a \in \mathbb{R}_- $ for $ y = 0 $ .

Solution to \eqref{eq:subgradient_1} might not exist, which implies that there might exist no support affine function of the convex extension with the slope $ w \in G, g_y \in \mathbb{R} $ where $ g_y $ is bounded, such that it is a convex lower bound on $ d_0 $ and $ d_1 $, and all of candidate minimizers in derived expressions either will not satisfy \eqref{eq:subgradient_1} or be undefined. Notice that when we set $ a = \pm \infty $ we get $ g_y = \pm \infty $, and any supporting affine function constructed using \eqref{eq:subgradient_1} with such $ g_y $ and $ w \in G' $ is a (vertical) supporting hyperplane as $ y' - y \leq 0 $, thus its slope defines an (unbounded) subgradient for $ y = 1 $.

Proof for the case where $ y = 0 $ is the same, except that we use different optimality condition 
\eqref{eq:minimizer_convex_envelope_equivalent} to avoid division by zero.



\end{document}

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\end{thebibliography}
