PoS is the true number of different solutions since it
finds all solutions for all nonnegative λ.
As shown in Table 2, Gurobi, with a low resolu-
tion (large ∆), fails to find some solutions that cor-
respond to λ not included in the list. With ∆ = 0.1,
both PoS and Gurobi have about the same compu-
tation time. However, Gurobi found only 24 differ-
ent solutions while PoS was able to produce all 114
different solutions. As ∆ decreases, the list of λ for
Gurobi becomes finer. Gurobi is able to find more
different solutions, but it also takes longer time. At
the resolution of 0.0005, Gurobi found 108 solutions,
which is close to the total number of different solu-
tions. However, it takes Gurobi more than 80 seconds
while PoS can achieve a better result in 0.42 seconds.
In Table 3 and Figure 7, we report the percent-
ages of solutions found by Gurobi compared to PoS,
averaged across 10 runs. A reported number of 100%
implies that we find all possible solutions. We also re-
port the average relative runtime of Gurobi compared
to PoS, where the reported number of a for Gurobi
implies the runtime that is a times of that of PoS.
When we set the resolution for Gurobi at 0.1,
Gurobi and PoS take approximately the same amount
of time. However, with this resolution, Gurobi only
found about 23% of all possible solutions while PoS
can find all of them. At the finest resolution included
in the experiment, which is 0.0005, Gurobi can find
97% of all solutions. However, it takes more than 200
times longer than PoS.
7 CONCLUSIONS
We provide in this work an efficient minimum cut-
based algorithm called Path of Solutions, or PoS, that
generates the solutions of the convex piecewise linear
fused lasso problem (1) for all values of the tradeoff
parameter, λ. PoS is a more efficient alternative to
the traditional method of parameter tuning, in which
a single parameter value is evaluated one at a time. In
traditional parameter tuning, we might unknowingly
overlook some important values or test two redundant
values that lead to the same result. These challenges
are overcome by PoS.
In addition to the algorithm design, we provide
both the time complexity of the algorithm and the
bound of the number of different solutions across all
nonnegative λ in terms of the number of variables and
the number of breakpoints in the fidelity functions.
We demonstrate the efficiency of PoS via a set of
experiments in comparison with Gurobi, a state-of-
the-art solver for mathematical programming. PoS
is capable of generating all solutions for all λ while
Gurobi found only a small fraction in the same
amount of time. To find all solutions, Gurobi requires
more than a factor of 200 of the time that PoS needs.
Regarding future directions of this work, we are
interested in extending our algorithm to solve other
variants of the fused lasso problems as well as con-
ducting more experiments on synthetic and real data.
ACKNOWLEDGEMENTS
This research was supported in part by AI Institute
NSF Award 2112533.
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