Learning Social Welfare Functions

TL;DR

It is shown that power mean functions are learnable with polynomial sample complexity in both cases, even if the comparisons are social welfare information is noisy.

Abstract

Is it possible to understand or imitate a policy maker's rationale by looking at past decisions they made? We formalize this question as the problem of learning social welfare functions belonging to the well-studied family of power mean functions. We focus on two learning tasks; in the first, the input is vectors of utilities of an action (decision or policy) for individuals in a group and their associated social welfare as judged by a policy maker, whereas in the second, the input is pairwise comparisons between the welfares associated with a given pair of utility vectors. We show that power mean functions are learnable with polynomial sample complexity in both cases, even if the comparisons are social welfare information is noisy. Finally, we design practical algorithms for these tasks and evaluate their performance.

Figures (8)

• Figure 1: Results for cardinal case with number of samples. Different lines show results for different values of added noise $\nu$. Solid lines correspond to values for learned parameters, whereas dotted lines correspond to values for real parameters.
• Figure 2: Results for ordinal case with number of samples. Different lines show results for different values of noise level $\tau$. Solid lines correspond to values for learned parameters, whereas dotted lines correspond to values for real parameters.
• Figure 3: More results for cardinal case with number of samples. Different lines show results for different values of added noise. Solid lines correspond to values for learnt parameters, whereas dotted lines correspond to values for real parameters.
• Figure 4: More results for ordinal case with number of samples. Different lines show results for different values of $\tau$. Solid lines correspond to values for learnt parameters, whereas dotted lines correspond to values for real parameters.
• Figure 5: More results for ordinal case with $p = 1.62$. Different lines show results for different values of $\tau$. Solid lines correspond to values for learnt parameters, whereas dotted lines correspond to values for real parameters.

Theorems & Definitions (31)

• Lemma 2.1
• Lemma 3.1
• Theorem 3.2
• Lemma 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 5.1
• proof : Proof of \ref{['prop:increasing_p_w']} \ref{['lem:monotone_a']}
• proof : Proof of \ref{['prop:increasing_p_w']} \ref{['lem:monotone_b']}
• proof : Proof of \ref{['prop:increasing_p_w']} \ref{['lem:quasilinear_c']}