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tschm merged 2 commits into from
Oct 24, 2026
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constrained_lasso: say where the CLA alone falls short #65

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constrained_lasso: say where the CLA alone falls short
tschm Oct 24, 2026
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constrained_lasso: name the machine behind the cost ratios
tschm Oct 24, 2026
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15 changes: 10 additions & 5 deletions preprints/constrained_lasso/constrained_lasso.tex
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Expand Up @@ -147,9 +147,12 @@ \section{A constrained regression}
as a budget, position limits and sector exposures, and Markowitz built them into the
CLA from the start. If the identity holds, a CLA written and tested for portfolios
should return the knots of a constrained LASSO, constraints included, with no change to
its code. This note checks that prediction. It was not quite met: the checks exposed
two defects in the CLA that also refuse ordinary portfolio problems, and they were
fixed in \texttt{cvxcla} 2.6.0, the release used here (Section~\ref{sec:limits}). The two encodings,
its code. This note checks that prediction. It was not quite met, in two ways. The
checks exposed two defects in the CLA that also refuse ordinary portfolio problems,
and they were fixed in \texttt{cvxcla} 2.6.0, the release used here. And where an
encoding starts degenerate, the CLA alone does not suffice: \texttt{cvxcla.Lasso} runs
it twice on a relaxed problem and combines the two paths, outside the CLA
(Section~\ref{sec:limits}). The two encodings,
Propositions~\ref{prop:penalty} and~\ref{prop:budget}, are proved here from the
definition of the CLA. They need nothing from \cite{perspective} beyond the classical
equivalence of the bound and penalty forms of the LASSO; that paper supplies the
Expand Down Expand Up @@ -377,7 +380,8 @@ \section{The penalty form}
\section{The budget form}
\label{sec:budget}

The penalty form cannot take a zero sum: with $A\beta=0$ its start is degenerate
The penalty form cannot take a zero sum directly: with $A\beta=0$ its start is
degenerate, and it reaches one only by relaxing the constraint
(Section~\ref{sec:limits}). The second encoding starts instead from Tibshirani's
original statement, the budget $\|\beta\|_1\le t$, and divides by $t$. The split
coefficients then add up to at most one, which is Markowitz's own setting: a long-only
Expand Down Expand Up @@ -593,7 +597,8 @@ \section{Numerical evidence}
through CVXPY, compiled once with $\lambda$ as a parameter) costs about 100 times as
much as the whole CLA path at $p=100$ and 350 times at $p=500$, and 14 to 100 times on
the constrained diabetes and zero-sum problems of Table~\ref{tab:evidence}, in ratios of
the fastest of several runs on one machine; \texttt{timings} measures them on another.
the fastest of several runs on one machine, an Apple M4 Pro; \texttt{timings} measures
them on another.
The grid also misses every knot between grid points.

\paragraph{Hostile problems} The problems so far are easy for any path algorithm.
Expand Down
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