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Algorithmic Challenges in Ensuring Fairness at the Time of Decision

Jad Salem, Swati Gupta, Vijay Kamble

TL;DR

Research shows that optimizing pricing trajectories, even under monotonicity constraints, does not compromise optimal regret guarantees for maximizing revenue and opens new avenues for how online platforms can rethink pricing strategies to foster consumer loyalty without sacrificing profitability.

Abstract

Algorithmic decision-making in societal contexts, such as retail pricing, loan administration, recommendations on online platforms, etc., can be framed as stochastic optimization under bandit feedback, which typically requires experimentation with different decisions for the sake of learning. Such experimentation often results in perceptions of unfairness among people impacted by these decisions; for instance, there have been several recent lawsuits accusing companies that deploy algorithmic pricing practices of price gouging. Motivated by the changing legal landscape surrounding algorithmic decision-making, we introduce the well-studied fairness notion of envy-freeness within the context of stochastic convex optimization. Our notion requires that upon receiving decisions in the present time, groups do not envy the decisions received by any of the other groups, both in the present as well as the past. This results in a novel trajectory-constrained stochastic optimization problem that renders existing techniques inapplicable. The main technical contribution of this work is to show problem settings where there is no gap in achievable regret (up to logarithmic factors) when envy-freeness is imposed. In particular, in our main result, we develop a near-optimal envy-free algorithm that achieves $\tilde{O}(\sqrt{T})$ regret for smooth convex functions that satisfy the PL inequality. This algorithm has a coordinate-descent structure, in which we carefully leverage gradient information to ensure monotonic sampling along each dimension, while avoiding overshooting the constrained optimum with high probability. This latter aspect critically uses smoothness and the structure of the envy-freeness constraints, while the PL inequality allows for sufficient progress towards the optimal solution. We discuss several open questions that arise from this analysis, which may be of independent interest.

Algorithmic Challenges in Ensuring Fairness at the Time of Decision

TL;DR

Research shows that optimizing pricing trajectories, even under monotonicity constraints, does not compromise optimal regret guarantees for maximizing revenue and opens new avenues for how online platforms can rethink pricing strategies to foster consumer loyalty without sacrificing profitability.

Abstract

Algorithmic decision-making in societal contexts, such as retail pricing, loan administration, recommendations on online platforms, etc., can be framed as stochastic optimization under bandit feedback, which typically requires experimentation with different decisions for the sake of learning. Such experimentation often results in perceptions of unfairness among people impacted by these decisions; for instance, there have been several recent lawsuits accusing companies that deploy algorithmic pricing practices of price gouging. Motivated by the changing legal landscape surrounding algorithmic decision-making, we introduce the well-studied fairness notion of envy-freeness within the context of stochastic convex optimization. Our notion requires that upon receiving decisions in the present time, groups do not envy the decisions received by any of the other groups, both in the present as well as the past. This results in a novel trajectory-constrained stochastic optimization problem that renders existing techniques inapplicable. The main technical contribution of this work is to show problem settings where there is no gap in achievable regret (up to logarithmic factors) when envy-freeness is imposed. In particular, in our main result, we develop a near-optimal envy-free algorithm that achieves regret for smooth convex functions that satisfy the PL inequality. This algorithm has a coordinate-descent structure, in which we carefully leverage gradient information to ensure monotonic sampling along each dimension, while avoiding overshooting the constrained optimum with high probability. This latter aspect critically uses smoothness and the structure of the envy-freeness constraints, while the PL inequality allows for sufficient progress towards the optimal solution. We discuss several open questions that arise from this analysis, which may be of independent interest.

Paper Structure

This paper contains 42 sections, 10 theorems, 51 equations, 9 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Modeling Envy-Freeness in Stochastic Convex Optimization
  3. Envy-Free Trajectory Constraints:
  4. Objective:
  5. Overview of results
  6. Noiseless Feedback.
  7. Noisy Feedback.
  8. Organization.
  9. Applications of EFTD
  10. Related Work
  11. Fairness in static decision-making.
  12. Temporal notions of fairness in dynamic decision-making.
  13. Regret-optimal EFTD algorithms without noise
  14. Algorithm design for N=1 group
  15. Algorithm design for N>1 groups
  16. ...and 27 more sections

Key Result

Lemma 4.1

Let $f : \mathcal{X} \to \mathbf{R}$ be a function satisfying Assumption asp:st-sm. Let $x_1,\ldots,x_{T/2}$ be the non-lagged points generated by Lagged Gradient Descent (Algorithm alg:Lagged-2pt-Gradient-Descent-Backtracking), and assume that $x_1 \leq x^* = \arg\,\min_{x \in \mathbb{R}} f(x)$. Th

Figures (9)

  • Figure 1: The price path of Purell hand sanitizer on Amazon from March 2015 to March 2021 (from https://camelcamelcamel.com/product/B00U2KYUAY).
  • Figure 2: (a) Example of a scenario in which monotonic coordinate descent does not converge to the constrained optimum $\mathbf{x}^*_C$. Here, $\mathbf{x}_t(1)$ cannot be optimized further, since it is at a tight constraint, and $\mathbf{x}_t(2)$ cannot be optimized further, since it is already at its optimum. Yet, $\mathbf{x}_t$ still has not attained its joint constrained optimum of $\mathbf{x}^*_C$. (b) Trajectory of the continuous-time algorithm.
  • Figure 3: Illustration of the points: the algorithm starts exploring at $x_{\min} = x_1 - \delta_1$ followed by $x_1 - \delta_2$. In this case, Phase 1 consists of $x_1$, Phase 2 consists of $x_2$ and $x_3$, Phase 3 consists of $x_4$, and Phase 4 consists of $x_5$; the step-size indices are $n_1= 2, n_2=5, n_3=8$, and $n_4=9$. The computation of $x_{t+1}$ is given by approximate gradient from the chosen lagged point, as depicted by the dotted lines, using the estimate $\widetilde{\widetilde{\nabla}}_t(x_t - \delta_i,x_t)$ obtained by sampling at $x_t-\delta_i$ and $x_t$.
  • Figure 4: An illustration of a potential decision trajectory of Algorithm \ref{['alg:noisy-2segment-asymmetric-simplified-v2']}, drawn over a shaded decision space EF. The red iterates are sampled to estimate $f_1'$, the blue iterates are sampled to estimate $f_2'$, points marked with an "x" are lagged iterates, brown vertices are the "feasibility check iterates" described in the algorithm, and pink iterates in the combined phase. Points which are infeasible when calculated are shaded and will be sampled if and when they become feasible.
  • Figure 5: (left) The cumulative number of non-monotonic steps taken by Dtu. (center) The cumulative number of non-monotonic steps (green) and the cumulative number of non-monotonic steps of size at least $0.02$ (blue) generated by hazan2014bandit. Both simulations were on input of data-derived revenue curves corrupted by independent $N(0,.4)$ noise over a price space of $[0,1]$. (right) Price path generated by Ada-Lgd (Alg. \ref{['alg:continual-lgd-convex']}).
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 1.1
  • Claim 1.1
  • Lemma 4.1
  • Theorem 4.1
  • Claim 4.1: continuous-time algorithm does not get stuck
  • Claim 4.2: no overshooting the constrained optimum
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • ...and 17 more