Prof Sarah Dean

Announcements

• Rank preferences for paper presentations
• Email me about first paper presentations 9/12 and 9/14
• HSNL18 Fairness Without Demographics in Repeated Loss Minimization
• PZMH20 Performative Prediction
• Required: meet with Atul at least 2 days before you are scheduled to present
• Working in pairs/groups, self-assessment

training data

$$\{(x_i, y_i)\}$$

model

$$f:\mathcal X\to\mathcal Y$$

policy

observation

action

ML in Feedback Systems

training data

$$\{(x_i, y_i)\}$$

model

$$f:\mathcal X\to\mathcal Y$$

observation

prediction

$$\mathcal D$$

sampled i.i.d. from $$\mathcal D$$

$$x\sim\mathcal D_{x}$$

Goal: for new sample $$x,y\sim \mathcal D$$, prediction $$\hat y = f(x)$$ is close to true $$y$$

Predictions via Risk Mimization

Goal: for new sample $$x,y\sim \mathcal D$$, prediction $$\hat y = f(x)$$ is close to true $$y$$

$$\ell(y,\hat y)$$  measures "loss" of predicting $$\hat y$$ when it's actually $$y$$

Encode our goal in risk minimization framework:

$$\min_{f\in\mathcal F}\mathcal R(f) = \mathbb E_{x,y\sim\mathcal D}[\ell(y, f(x))]$$

$$\hat \theta = \arg\min \sum_{i=1}^N(-\theta^\top x_i\cdot y_i)_+$$

predict $$\hat f(x) = \mathbb 1\{\hat\theta^\top x \geq t\}$$

No fairness through unawareness!

$$x_i=$$ demographic info and browsing history

$$y_i=$$ clicked (1) or not (-1)

The index of $$\hat\theta$$ corresponding to "female" is negative!

The index of $$\hat\theta$$ corresponding to "visited website for women's clothing store" is negative!

Statistical Classification Criteria

Accuracy
$$\mathbb P( \hat Y = Y)$$ = ________

Positive rate
$$\mathbb P( \hat Y = 1)$$ = ________

False positive rate
$$\mathbb P( \hat Y = 1\mid Y = 0)$$ = ________

False negative rate

$$\mathbb P( \hat Y = 0\mid Y = 1)$$ = ________

Positive predictive value
$$\mathbb P( Y = 1\mid\hat Y = 1)$$ = ________

Negative predictive value
$$\mathbb P( Y = 0\mid\hat Y = 0)$$ = ________

$$X$$

$$Y=1$$

$$Y=0$$

$$f(X)$$

$$3/4$$

$$9/20$$

$$1/5$$

$$3/10$$

$$7/9$$

$$8/11$$

Non-discrimination Criteria

• Informally: predictor should treat individuals the "same" across groups
• Formally: equalize positive rate, error rate, or predictive value across groups

Independence: prediction does not depend on $$a$$

$$\hat y \perp a$$

e.g. ad displayed at equal rates across gender

Separation: given outcome, prediction does not depend on $$a$$

$$\hat y \perp a~\mid~y$$

e.g. ad displayed to interested users at equal rates across gender

Sufficiency: given prediction, outcome does not depend on $$a$$

$$y \perp a~\mid~\hat y$$

e.g. users viewing ad are interested at equal rates across gender

In addition to features $$x$$ and labels $$y$$, individuals have protected attribute $$a$$ (e.g. gender, race)

Ref: Ch 2 of Hardt & Recht, "Patterns, Predictions, and Actions" mlstory.org; Ch 3 of Barocas, Hardt, Narayanan "Fairness and Machine Learning" fairmlbook.org.

Ex: Pre-trial Detention

• COMPAS: a criminal risk assessment tool used in pretrial release decisions
• $$x$$ survey about defendant
• $$\hat y$$ designation as high- or low-risk
• Audit of data from Broward county, FL
• $$a$$ race of defendant
• $$y$$ recidivism within two years

“Black defendants who did not recidivate over a two-year period were nearly twice as likely to be misclassified. [...] White defendants who re-offended within the next two years were mistakenly labeled low risk almost twice as often.”

Ex: Pre-trial Detention

“In comparison with whites, a slightly lower percentage of blacks were ‘Labeled Higher Risk, But Didn’t Re-Offend.’ [...] A slightly higher percentage of blacks were ‘Labeled Lower Risk, Yet Did Re-Offend.”’

$$\mathbb P(\hat y = 1\mid y=0, a=\text{Black})> \mathbb P(\hat y = 1\mid y=0, a=\text{White})$$

$$\mathbb P(\hat y = 0\mid y=1, a=\text{Black})< \mathbb P(\hat y = 0\mid y=1, a=\text{White})$$

$$\mathbb P(y = 0\mid \hat y=1, a=\text{Black})\approx \mathbb P( y = 0\mid \hat y=1, a=\text{White})$$

$$\mathbb P(y = 1\mid \hat y=0, a=\text{Black})\approx \mathbb P( y = 1\mid \hat y=0, a=\text{White})$$

COMPAS risk predictions do not satisfy separation

Ex: Pre-trial Detention

COMPAS risk predictions do satisfy sufficiency

Achieving Nondiscrimination Criteria

• Pre-processing: remove correlations between $$a$$ and features $$x$$ in dataset.
• Pre-processing: remove correlations between $$a$$ and features $$x$$ in dataset.
• Requires knowledge of $$a$$ during data cleaning

Achieving Nondiscrimination Criteria

• Pre-processing: remove correlations between $$a$$ and features $$x$$ in dataset.
• In-processing: modify learning algorithm to respect criteria.
• Requires knowledge of $$a$$ at training time

Achieving Nondiscrimination Criteria

• Pre-processing: remove correlations between $$a$$ and features $$x$$ in dataset.
• In-processing: modify learning algorithm to respect criteria.
• Post-processing: adjust thresholds in group-dependent manner.

Achieving Nondiscrimination Criteria

• Pre-processing: remove correlations between $$a$$ and features $$x$$ in dataset.
• In-processing: modify learning algorithm to respect criteria.
• Post-processing: adjust thresholds in group-dependent manner.
• Requires knowledge of $$a$$ at decision time

Limitations of Nondiscrimination Criteria

• Tradeoffs:  It is impossible to simultaneously satisfy separation and sufficiency if populations have different base rates

Limitations of Nondiscrimination Criteria

• Tradeoffs:  It is impossible to simultaneously satisfy separation and sufficiency if populations have different base rates
• Observational: Statistical criteria can measure only correlation; intuitive notions of discrimination involve causation, which requires careful modelling

Limitations of Nondiscrimination Criteria

• Tradeoffs:  It is impossible to simultaneously satisfy separation and sufficiency if populations have different base rates
• Observational: Statistical criteria can measure only correlation; intuitive notions of discrimination involve causation, which requires careful modelling
• Unclear legal grounding: While algorithmic decisions may have disparate impact, achieving criteria involves disparate treatment

Barocas & Selbst, Big Data's Disparate Impact

Limitations of Nondiscrimination Criteria

• Tradeoffs:  It is impossible to simultaneously satisfy separation and sufficiency if populations have different base rates
• Observational: Statistical criteria can measure only correlation; intuitive notions of discrimination involve causation, which requires careful modelling
• Unclear legal grounding: While algorithmic decisions may have disparate impact, achieving criteria involves disparate treatment
• Limited view: focusing on risk prediction might miss the bigger picture of how these tools are used by larger systems to make decisisons

Discrimination Beyond Classification

image cropping

facial recognition

information retrieval

generative models

Empirical Risk Minimization

1. define loss
2. do ERM

performance depends on risk $$\mathcal R(f)$$

$$\hat f = \min_{f\in\mathcal F} \frac{1}{n} \sum_{i=1}^n \ell(y_i, f(x_i))$$

$$\{$$

$$\mathcal R_N(f)$$

(with fairness constraints)

training data

$$\{(x_i, y_i)\}$$

model

$$f:\mathcal X\to\mathcal Y$$

Sample vs. population

Fundamental Theorem of Supervised Learning:

• The risk is bounded by the empirical risk plus the generalization error. $$\mathcal R(f) \leq \mathcal R_N(f) + |\mathcal R(f) - \mathcal R_N(f)|$$

Empirical risk minimization

$$\hat f = \min_{f\in\mathcal F} \frac{1}{n} \sum_{i=1}^n \ell(y_i, f(x_i))$$

$$\{$$

$$\mathcal R_N(f)$$

1. Representation

2. Optimization

3. Generalization

Case study: linear regression

At first glance, linear representation seems limiting

Least-squares linear regression models $$y\approx \theta^\top x$$

$$\min_{\theta\in\mathbb R^d} \frac{1}{n}\sum_{i=1}^n \left(\theta^\top x_i - y_i\right)^2$$

but we can encode rich representations by expanding the features (increasing $$d$$)

$$y = (x-1)^2$$

$$y = \begin{bmatrix}1\\-2\\1\end{bmatrix}^\top \begin{bmatrix}1\\x\\x^2\end{bmatrix}$$

$$\varphi(x)$$

$$\{$$

For more, see Ch 4 of Hardt & Recht, "Patterns, Predictions, and Actions" mlstory.org.

Case study: linear regression

one global min

infinitely many global min

local and global min

Optimization is straightforward due to differentiable and convex risk

$$\min_{\theta\in\mathbb R^d} \frac{1}{n}\sum_{i=1}^n \left(\theta^\top x_i - y_i\right)^2$$

strongly convex

convex

nonconvex

Case study: linear regression

Derivation of optimal solution

$$\hat\theta\in\arg\min_{\theta\in\mathbb R^d} \frac{1}{n}\sum_{i=1}^n \left(\theta^\top x_i - y_i\right)^2$$

first order optimality condition: $$\displaystyle \sum_{i=1}^n x_i x_i^\top\hat \theta = \sum_{i=1}^n y_ix_i$$

min-norm solution: $$\displaystyle \hat\theta = \left( \sum_{i=1}^n x_i x_i^\top\right)^\dagger\sum_{i=1}^n y_ix_i$$

Case study: linear regression

Proof: exercise. Hint: consider the span of the $$x_i$$.

Iterative optimization with gradient descent $$\theta_{t+1} = \theta_t - \alpha\sum_{i=1}^n (\theta_t^\top x_i - y_i)x_i$$

Claim: suppose $$\theta_0=0$$ and GD converges to a minimizer. Then it converges to the minimum norm solution.

Case study: linear regression

Generalization: under the fixed design generative model, $$\{x_i\}_{i=1}^n$$ are fixed and

$$y_i = \theta_\star^\top x_i + v_i$$ with $$v_i$$ i.i.d. with mean $$0$$ and variance $$\sigma^2$$

$$\mathcal R(\theta) = \frac{1}{n}\sum_{i=1}^n \mathbb E_{y_i}\left[(x_i^\top \theta - y_i)^2\right]$$

Claim: when features span $$\mathbb R^d$$, the excess risk $$\mathcal R(\hat\theta) -\mathcal R(\theta_\star) =\frac{\sigma^2 d}{n}$$

• First, $$\mathcal R(\theta) = \frac{1}{n}\|X(\theta-\theta_\star)\|_2^2 + \sigma^2$$
• Then, $$\|X(\hat\theta-\theta_\star)\|_2^2 = v^\top X(X^\top X)^{-1}X^\top v$$
• Then take expectation.

Case study: linear regression

Exercises:

• For the same generative model and a new fixed $$x_{n+1}$$, what is the expected loss $$\mathbb E_y[(\hat \theta^\top x_{n+1} - y_{n+1})^2]$$? Can you interpret the quantities?
• In the random design setting, we take each $$x_i$$ to be drawn i.i.d. from $$\mathcal N(0,\Sigma)$$ and assume that $$v_i$$ is also Gaussian. The risk is then $$\mathcal R(\theta) = \mathbb E_{x, y}\left[(x^\top \theta - y)^2\right].$$ What is the excess risk of $$\hat\theta$$ in terms of $$X^\top X$$ and $$\Sigma$$? What is the excess risk in terms of $$\sigma^2, n, d$$ (ref 1, 2)?

Recap

• Non-discrimination criteria
• independence, separation, and sufficiency
• Least-squares regression
• representation, optimization, generalization

Next time: online learning with linear least-squares case study

By Sarah Dean

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