Natural Analysts in Adaptive Data Analysis Tijana Zrnic joint with - - PowerPoint PPT Presentation

Natural Analysts in Adaptive Data Analysis

Tijana Zrnic
 joint with Moritz Hardt

Adaptivity in Machine Learning

Adaptivity in Machine Learning

data analyst
 with training data

Adaptivity in Machine Learning

data analyst
 with training data

model

Adaptivity in Machine Learning

data analyst
 with training data

test data set model

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

tune hyperparams...

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

tune hyperparams...

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

Adaptivity in Machine Learning

data analyst
 with training data

test data set model accuracy

tune hyperparams...

Adaptivity in Machine Learning

• After tested models, how well does the final model

generalize?


• Depends on how the accuracies are computed

t

data analyst
 with training data

test data set model accuracy

Classical Holdout vs Response Mechanism

data analyst
 with training data

test data set model accuracy

• Reporting exact sample accuracy allows for great overfitting

Classical Holdout vs Response Mechanism

data analyst
 with training data

test data set model accuracy

• Reporting exact sample accuracy allows for great overfitting
• Better bounds can be obtained by having a non-trivial response mechanism


in charge of reporting accuracy on the test data response mechanism
 for reporting accuracy

Classical Holdout vs Response Mechanism

data analyst
 with training data

test data set model accuracy

Main Questions

data analyst
 with training data

test data set model accuracy

response mechanism
 for reporting accuracy

Main Questions

data analyst
 with training data

test data set model accuracy

response mechanism
 for reporting accuracy

Main Questions

• How do we construct a mechanism such that its responses generalize to the

population?

data analyst
 with training data

test data set model accuracy

response mechanism
 for reporting accuracy

Main Questions

• How do we construct a mechanism such that its responses generalize to the

population?

• want 95% reported accuracy on test data ≈ 95% accuracy on fresh data from same population


data analyst
 with training data

test data set model accuracy

response mechanism
 for reporting accuracy

Main Questions

• How do we construct a mechanism such that its responses generalize to the

population?

• want 95% reported accuracy on test data ≈ 95% accuracy on fresh data from same population

• For such a good mechanism, how much does a possibly adversarial analyst overfit?

Abstraction via Adaptive Data Analysis

Abstraction via Adaptive Data Analysis

Framework of Dwork et al. (2015)

Abstraction via Adaptive Data Analysis

analyst Framework of Dwork et al. (2015)

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015)

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set S

S ∼ Pn

n

• sample size

P

• population distribution

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set

q1

query

S

S ∼ Pn

n

• sample size

P

• population distribution
• queries posed by analyst

qi : supp(P) → [0, 1]d

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set

q1

a1

query answer

S

S ∼ Pn

n

• sample size

P

• population distribution
• queries posed by analyst
• answers given by mechanism

qi : supp(P) → [0, 1]d ai ∈ Rd

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set

q1

a1

q2

query query answer

S

S ∼ Pn

n

• sample size

P

• population distribution
• queries posed by analyst
• answers given by mechanism

qi : supp(P) → [0, 1]d ai ∈ Rd

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set

q1

a1

q2

a2

query query answer answer

S

S ∼ Pn

n

• sample size

P

• population distribution
• queries posed by analyst
• answers given by mechanism

qi : supp(P) → [0, 1]d ai ∈ Rd

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set

q1

a1

q2

a2

query query answer answer

S

S ∼ Pn

n

• sample size

P

• population distribution
• queries posed by analyst
• answers given by mechanism

qi : supp(P) → [0, 1]d ai ∈ Rd

Abstraction via Adaptive Data Analysis

analyst mechanism Framework of Dwork et al. (2015) data set

q1

a1

q2

a2

query query answer answer

S

S ∼ Pn

n

• sample size

P

• population distribution
• queries posed by analyst
• answers given by mechanism

S = {X1, . . . , Xn}

• data set

qi : supp(P) → [0, 1]d ai ∈ Rd

Generalization Error

analyst Framework of Dwork et al. (2015)

q1

a1

q2

a2

query query answer answer mechanism data set S

S ∼ Pn

Goal: design a mechanism such that, for queries posed by the analyst, generalization error is at most :

Generalization Error

t

✏

max

1≤i≤t kEX∼P[qi(X)] aik∞  ✏

with high probability.

analyst Framework of Dwork et al. (2015)

q1

a1

q2

a2

query query answer answer mechanism data set S

S ∼ Pn

examples of analysts human
 analyst iterative algorithms, e.g. gradient descent

• classification

error of


• th classifier on

data set

• gradient of 


the empirical risk

• n data set

qi

i

S

qi

S

examples of mechanisms empirical mechanism Gaussian mechanism truncation to a fixed number

• f bits

ai = 1 n

n

• j=1

qi(Xj)

ai = trunc

• 1

n

n

• j=1

qi(Xj)

• ai = 1

n

n

• j=1

qi(Xj) + ξi, ξi ∼ N(0, σ2Id)

Known bounds on the generalization error after queries: t

non-adaptive
 analyst fully adaptive
 analyst

Known bounds on the generalization error after queries: t

non-adaptive
 analyst fully adaptive
 analyst generalization error generalization error

Known bounds on the generalization error after queries: t

O

• log(td)

n

• ˜

O (td)1/4 √n

non-adaptive
 analyst fully adaptive
 analyst generalization error generalization error

Known bounds on the generalization error after queries: t

tight tight*

O

• log(td)

n

• ˜

O (td)1/4 √n

• *tight for a broad class of mechanisms, believed to be tight in general

non-adaptive
 analyst fully adaptive
 analyst generalization error generalization error

Known bounds on the generalization error after queries: t

tight tight*

• verly optimistic
• verly pessimistic

O

• log(td)

n

• ˜

O (td)1/4 √n

• *tight for a broad class of mechanisms, believed to be tight in general

non-adaptive
 analyst fully adaptive
 analyst generalization error generalization error

Known bounds on the generalization error after queries: t

• verly optimistic

natural analysts?

Are there natural categories of analysts which interpolate between logarithmic and polynomial error?

• verly pessimistic

O

• log(td)

n

• ˜

O (td)1/4 √n

Analysts as Dynamical Systems

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

a2

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

a2

h3

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

a2

h3

q3

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

a2

h3

q3

a3

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

a2

h3

q3

a3

Analysts as Dynamical Systems

We model the data analyst as a dynamical system:

ht = ψt(ht−1, at−1) qt = ft(ht)

ψt

ft

ht

• history, i.e. encoding of past interactions
• arbitrary transition map
• arbitrary function

h1

q1

a1

h2

q2

a2

h3

q3

a3

With no restriction on the transition map, this representation captures an arbitrary adaptive analyst

Stability of Natural Analysts

Stability of Natural Analysts

• Stability of dynamical systems makes analysts natural

Stability of Natural Analysts

• Stability of dynamical systems makes analysts natural
• Stability ensures convergence of algorithms, encodes

different human biases, like sensitivity to interactions far enough in the past, etc.


Stability of Natural Analysts

• Stability of dynamical systems makes analysts natural
• Stability ensures convergence of algorithms, encodes

different human biases, like sensitivity to interactions far enough in the past, etc.


• Encoding different stability notions, we introduce two

main classes of natural analysts: progressive and conservative


Progressive Analysts

Progressive Analysts

Progressive analysts contract their history as: kψt(h, a) ψt(h0, a)k  λkh h0k, 8h, h0, a

for some λ ∈ (0, 1).

The parameter encodes how fast past interactions with the mechanism are forgotten; is minimal adaptivity, while implies full adaptivity.

Progressive Analysts

Progressive analysts contract their history as: kψt(h, a) ψt(h0, a)k  λkh h0k, 8h, h0, a

for some λ ∈ (0, 1).

λ

λ ≈ 0 λ ≈ 1

The parameter encodes how fast past interactions with the mechanism are forgotten; is minimal adaptivity, while implies full adaptivity.

Progressive Analysts

Progressive analysts contract their history as:

examples of progressive analysts human
 analysts algorithmic analysts analysts with recency bias stable RNNs, Bellman

• perator

kψt(h, a) ψt(h0, a)k  λkh h0k, 8h, h0, a

for some λ ∈ (0, 1).

λ

λ ≈ 0 λ ≈ 1

Main Theorem for Progressive Analysts

Main Theorem for Progressive Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a progressive analyst such that the overall generalization error is at most t

≈ ˜ O

• log(1/(1 − λ)) log(t)d

log(1/λ)n

Main Theorem for Progressive Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a progressive analyst such that the overall generalization error is at most t

≈ ˜ O

• log(1/(1 − λ)) log(t)d

log(1/λ)n

Main Theorem for Progressive Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a progressive analyst such that the overall generalization error is at most t

For the analyst is fully adaptive and we recover a (suboptimal) fully adaptive bound

λ = 1 − 1/t

≈ ˜ O

• log(1/(1 − λ)) log(t)d

log(1/λ)n

• ˜

O(

• td/n)

Main Theorem for Progressive Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a progressive analyst such that the overall generalization error is at most t

For the analyst is fully adaptive and we recover a (suboptimal) fully adaptive bound

λ = 1 − 1/t

For the analyst can only adapt to the last answer and we have

λ = 0

≈ ˜ O

• log(1/(1 − λ)) log(t)d

log(1/λ)n

• ˜

O(

• td/n)

˜ O(

• log(t)d/n)

Conservative Analysts

Conservative Analysts

Conservative analysts contract new evidence* as: kψt(h, a) ψt(h, a0)k  ηtka a0k, 8h, a, a0

for some sequence such that {ηt}

lim

t→∞ ηt = 0.

*alternate condition for conservative analysts given in paper

Conservative Analysts

Conservative analysts contract new evidence* as: kψt(h, a) ψt(h, a0)k  ηtka a0k, 8h, a, a0

for some sequence such that {ηt}

lim

t→∞ ηt = 0.

The sequence encodes how fast the knowledge of the analyst saturates. {ηt}

*alternate condition for conservative analysts given in paper

Conservative Analysts

Conservative analysts contract new evidence* as:

examples of conservative analysts human
 analysts algorithmic analysts analysts with anchoring bias

• ptimization

algorithms, e.g. gradient descent

kψt(h, a) ψt(h, a0)k  ηtka a0k, 8h, a, a0

for some sequence such that {ηt}

lim

t→∞ ηt = 0.

The sequence encodes how fast the knowledge of the analyst saturates. {ηt}

*alternate condition for conservative analysts given in paper

Main Theorem for Conservative Analysts

Main Theorem for Conservative Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a conservative analyst such that the overall generalization error is at most t

for some constant C.

≈ ˜ O (min{t, K(ηt)}d log(t))1/4 √n

• , K(ηt) = min{t : ηt ≤ C/

√ d}

Main Theorem for Conservative Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a conservative analyst such that the overall generalization error is at most t

for some constant C.

If we recover the non-adaptive bound ηt ≈ 0, ∀t

≈ ˜ O (min{t, K(ηt)}d log(t))1/4 √n

• , K(ηt) = min{t : ηt ≤ C/

√ d}

˜ O(

• log(td)/n)

Main Theorem for Conservative Analysts

There exists a computationally efficient mechanism for answering queries chosen adaptively by a conservative analyst such that the overall generalization error is at most t

for some constant C.

If we recover the non-adaptive bound ηt ≈ 0, ∀t

If has a slow decay, we recover the (tight) bound under full adaptivity

{ηt}

≈ ˜ O (min{t, K(ηt)}d log(t))1/4 √n

• , K(ηt) = min{t : ηt ≤ C/

√ d}

˜ O(

• log(td)/n)

˜ O((td)1/4/√n)

Summary

• Generalization bounds in adaptive data analysis show a

wide gap due to considering only overly optimistic or

• verly pessimistic settings

• In our work, we smoothly interpolate between the two by

using stability parameters as a knob

Future Directions

• Empirical evaluation of patterns of human adaptivity
• Preventing the analyst from knowing the distribution of

the data

• Limiting query complexity

Thank you