Preference and Participation Dynamics in Learning Systems

Sarah Dean, Cornell University

4th Annual Learning for Dynamics & Control Conference, June 2022

Large-scale automated systems

enabled by machine learning

\(\to\)

historical movie ratings

new movie rating

from actions impacting the world

Dynamics arise

from data impacting the policy

Outline

1. Preference dynamics

2. Participation dynamics

Preference Dynamics

\(u_t\)

\(y_t = g_t(x_t, u_t) \)

Interests may be impacted by recommended content

\(x_{t+1} = f_t(x_t, u_t)\)

expressed preferences

recommended content

recommender policy

Preference Dynamics

\(u_t\)

\(y_t = \langle x_t, u_t\rangle + w_t  \)

Interests may be impacted by recommended content

\(x_{t+1} = f_t(x_t, u_t)\)

expressed preferences

recommended content

recommender policy

underlies factorization-based methods

Preference Dynamics

\(u_t\)

\(y_t = \langle x_t, u_t\rangle + w_t  \)

expressed preferences

recommended content

recommender policy

underlies factorization-based methods

A model inspired by biased assimilation updates proportional to affinity

\(x_{t+1} \propto x_t + \eta_t\langle x_t, u_t\rangle u_t\)

Preference Dynamics

items \(u_t\in\mathcal U\subseteq \mathcal S^{d-1}\)

\(y_t = \langle x_t, u_t\rangle + w_t  \)

A model inspired by biased assimilation updates proportional to affinity

\(x_{t+1} \propto x_t + \eta_t\langle x_t, u_t\rangle u_t\)

preferences \(x\in\mathcal S^{d-1}\)

Proposed by Hązła et al. (2019) as model of opinion dynamics

Preference Dynamics

\(y_t = \langle x_t, u_t\rangle + w_t  \)

A model inspired by biased assimilation updates proportional to affinity

\(x_{t+1} \propto x_t + \eta_t\langle x_t, u_t\rangle u_t\)

Non-personalized exposure leads to polarization (Hązła et al. 2019; Gaitonde et al. 2021)

Preference Dynamics

\(y_t = \langle x_t, u_t\rangle + w_t  \)

A model inspired by biased assimilation updates proportional to affinity

\(x_{t+1} \propto x_t + \eta_t\langle x_t, u_t\rangle u_t\)

Personalized fixed recommendation \(u_t=u\)

$$ x_t = \alpha_t x_0 +  \beta_t u$$

positive and decreasing

increasing magnitude, same sign as \(\langle x_0, u\rangle\)

Affinity Maximization

regret of fixed strategy

Result: As long as \(|\langle x_0, u\rangle| > c\) and noise is \(\sigma^2\) sub-Gaussian, $$R(T)=  \sum_{t=0}^{T-1} 1 - \langle x_t, u_t \rangle \leq C_\eta(1/c^2 - 1) + \sigma^2\log T/c^2$$

Alg: Explore-then-Commit

  • For \(t=0,\dots,\sigma^2 \log T/c^2\):
    • recommend \(u_t=u\) and observe \(y_t\)
  • For \(t=\sigma^2 \log T/c^2,\dots,T\):
    • if \(\sum_t y_t < 0\): choose \(u_t=-u\)
    • else: recommend \(u_t=u\)

maximum possible affinity

regret of explore-then-commit

Achieving high affinity is straightforward when \(\mathcal U\) contains "opposites"

Non-manipulation (Krueger et al., 2020) is an alternative goal

$$R(T) = \sum_{t=0}^{T-1} 1 - \langle x_0, x_t \rangle $$

Preference Stationarity

When \(x_0\notin \mathcal U\), use randomized strategy to select \(u_t\) i.i.d.

$$\mathbb E[x_{t+1}] \propto (I+\eta_t\mathbb E[uu^\top])x_t$$

Informal Result: Suppose \(x_0\) is the dominant eigenvector of \(\mathbb E[uu^\top]\) and step size \(\eta_t \) decays like \(\frac{1}{1+t}\). Then

$$R(T) \lesssim \log T $$

Proof sketch:

$$\langle x_0 ,x_t\rangle = \frac{x_0^\top (I+u_{t-1}u_{t-1}^\top)\dots(I+u_{0}u_{0}^\top)x_0}{\|(I+u_{t-1}u_{t-1}^\top)\dots(I+u_{0}u_{0}^\top) x_0\|_2}$$

Using concentration for matrix products (Huang et al., 2021),

$$1-\langle x_0 ,x_t\rangle^2 \lesssim \frac{1}{t}$$

Mode Collapse

Rather than polarization....

...preferences may "collapse"

but this can be avoided using randomization

Necessary to have \(x_0\in\text{span}(\mathcal U)\) for

  1. estimating initial preference from observations
  2. designing stationary randomization

Observation function \(F(x_0; u_{0:T}) = y_{0:T}\) where \(y_t = \langle x_t, u_t\rangle\).

Result: \(F:\mathcal S^{d-1}\to \R^T\) is locally invertible if and only if \(u_{0:T}\) span \(\mathbb R^d\).

Randomization Design

Preference Estimation

find  \(q\)  such that  \(q\geq 0\),  \(U\mathrm{diag}(U^\top x_0) q = x_0\),

\(I-U\mathrm{diag}(q)U^\top\succeq 0\)  

Result: \(x_0\) is dominant eigenvector if randomization is proportional to \(q\).

Result: Problem is feasible if and only if \(x_0\) is in the span of \(\tilde \mathcal U = \{\mathrm{sign}(u^\top x_0)\cdot u\mid u\in \mathcal U\}\)

Open questions:

  • SysID for biased assimilation dynamics
  • Incorporate other phenomena (e.g. satiation, inconsistencies)

Preference Dynamics Summary

Key points:

  • Preferences with biased assimilation
  • High ratings vs. stationary preferences
  • Mode collapse rather than polarization

Outline

1. Preference dynamics

2. Participation dynamics

Does social media have the ability to manipulate us, or merely to segment and target?

For single learner, leads to representation disparity (Hashimoto et al., 2018; Zhang et al., 2019)

Choose to participate depending on accuracy (e.g. music recommendation)

Participation Dynamics

Self-reinforcing feedback loop when learners retrain

Sub-populations \(i\in[1,n]\)

Participation Dynamics

Learners \(j\in[1,m]\)

  • parameters \(\Theta = (\theta_1,\dots,\theta_m)\)


 

  • each learner \(j\) reduces observed (average) risk
  • participation \(\alpha\in\mathbb R^{n\times m}\)


 

  • each sub-pop \(i\) favors learners with lower risk

\(\alpha^{t+1} = \nu(\alpha^t, \Theta^t)\)

\(\Theta^{t+1} = \mu(\alpha^{t+1}, \Theta^t)\)

evolve according to risks \(\mathcal R_i (\theta_j)\)

"risk minimizing in the limit"

strongly convex

Example: linear regression with

  • sub-populations      and         
  • learners 1 and 2

2

1

Result: An equilibrium \((\alpha^{eq}, \Theta^{eq})\) must have \(\Theta^{eq}=\arg\min \mathcal R(\alpha^{eq},\Theta)\) and is asymptotically stable if and only if

  • \(\alpha^{eq}\) corresponds to split market
  • and \(\mathcal R_i(\theta_{\gamma(i)})< \mathcal R_i(\theta_j)~~\forall~~i,j\)

Definition: The total risk is   \(\mathcal R(\alpha,\Theta) = \sum_{i=1}^n \sum_{j=1}^m \alpha_{ij} \mathcal R_i(\theta_j)\)

Proof sketch: asympototically stable equilibria correspond to the isolated local minima of \(\mathcal R(\alpha,\Theta)\)

Definition: In a split market, each sub-pop \(i\) allocates all participation to a single learner \(\gamma(i)\)

Participation Equilibria

  • ✅ Stable equilibria (locally) maximize social welfare
  • ❌ Generally NP hard to find global max by analogy to clustering (Aloise et al., 2009)
  • ❌ Utilitarian welfare does not guarantee low worst-case risk
  • ✅ Increasing the number of learners decreases risk and increases social welfare

Utilitarian social welfare is inversely related to total risk

A notion of fairness is the worst-case risk over sub-pops

Welfare and Fairness Implications

Participation Dynamics Summary

Open questions:

  • Finite sample effects
  • Alternative participation dynamics
  • Alternative learning dynamics (e.g. DRO or competitive)

Key points:

  • Myopic risk reduction (participation and learning)
  • Segregated equilibria with locally maximized social welfare
  • Importance of multiple learners

Conclusion

Study dynamics to what end?

  • diagnostics
  • algorithm design
  • safety verification

How to bridge the social and the technical? (Gilbert et al., 2022)

  1. Preference Dynamics Under Personalized Recommendations (https://arxiv.org/abs/2205.13026) with Jamie Morgenstern
  2. Multi-learner risk reduction under endogenous participation dynamics (https://arxiv.org/abs/2206.02667) with Mihaela Curmei, Lillian J. Ratliff, Jamie Morgenstern, Maryam Fazel

References

  • Aloise, Deshpande, Hansen, Popat, 2009. NP-hardness of Euclidean sum-of-squares clustering. Machine learning.
  • Gaitonde, Kleinberg, Tardos, 2021. Polarization in geometric opinion dynamics. EC.
  • Gilbert, Dean, Lambert, Zick, Snoswell, 2022. Reward Reports for Reinforcement Learning. arXiv:2204.10817.
  • Hashimoto, Srivastava, Namkoong, Liang, 2018. Fairness without demographics in repeated loss minimization. ICML.
  • Hązła, Jin, Mossel, Ramnarayan, 2019. A geometric model of opinion polarization. arXiv:1910.05274.
  • Huang, Niles-Weed, Ward, 2021. Streaming k-PCA: Efficient guarantees for Oja’s algorithm, beyond rank-one updates. COLT.
  • Krueger, Maharaj, Leike, 2020. Hidden incentives for auto-induced distributional shift. arXiv:2009.09153.

  • Zhang, Khaliligarekani, Tekin, Liu, 2019. Group retention when using machine learning in sequential decision making: the interplay between user dynamics and fairness. NeurIPS.

Thanks! Questions?

Preference and Participation Dynamics in Learning Systems

By Sarah Dean

Preference and Participation Dynamics in Learning Systems

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