[PPT] - Learning wit ith Pairw rwis ise Losses Problems, Algorithms and PowerPoint Presentation

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Learning wit ith Pairw rwis ise Losses

Problems, Algorithms and Analysis

Purushottam Kar

Microsoft Research India

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Outl tline

Part I: Introduction to pairwise loss functions
Example applications
Part II: Batch learning with pairwise loss functions
Learning formulation: no algorithmic details
Generalization bounds
The coupling phenomenon
Decoupling techniques
Part III: Online learning with pairwise loss functions
A generic online algorithm
Regret analysis
Online-to-batch conversion bounds
A decoupling technique for online-to-batch conversions

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Part I: I: In Introduction

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What is is a lo loss fu functio ion?

We observe empirical losses on data 𝑇 = 𝑦1, … 𝑦𝑜

ℓ𝑦𝑗 ⋅ = ℓ ℎ, 𝑦𝑗

… and try to minimize them (e.g. classfn, regression)

ℎ = inf

ℎ∈ℋ

ℒ𝑇 ℎ , ℒ𝑇 ℎ = 1 𝑜 ∑ℓ𝑦𝑗 ℎ

… in the hope that

1 𝑜 ∑ℓ𝑦𝑗 ⋅ − 𝔽ℓ𝑦 ⋅

∞ ≤ 𝜗

... so that

ℒ ℎ ≤ ℒ ℎ∗ + 𝜗, ℒ ℎ = 𝔽ℓ𝑦 ℎ

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ℓ: ℋ → ℝ+

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Metric ic Learnin ing

Penalize metric for bringing blue and red points close
Loss function needs to consider two points at a time!
… in other words a pairwise loss function
E.g. ℓ 𝑦1,𝑦2

𝑁 = 1, 𝑧1 ≠ 𝑧2 and 𝑁 𝑦1, 𝑦2 < 𝛿1 1, 𝑧1 = 𝑧2 and 𝑁 𝑦1, 𝑦2 > 𝛿2 0, otherwise

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Pairw irwis ise Loss Functio ions

Typically, loss functions are based on ground truth

ℓ𝑦 ℎ = ℓ ℎ 𝑦 , 𝑧 𝑦

Thus, for metric learning, loss functions look like

ℓ 𝑦1,𝑦2 ℎ = ℓ ℎ 𝑦1, 𝑦2 , 𝑧 𝑦1, 𝑦2

In previous example, we had

ℎ 𝑦1, 𝑦2 = 𝑁 𝑦1, 𝑦2 and 𝑧 𝑦1, 𝑦2 = 𝑧1𝑧2

Useful to learn patterns that capture data interactions

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Pairw irwis ise Loss Functio ions

Examples: (𝜚 is any margin loss function e.g. hinge loss)

Metric learning [Jin et al NIPS ‘09]

ℓ 𝑦1,𝑦2 𝑁 = 𝜚 𝑧1𝑧2 1 − 𝑁 𝑦1, 𝑦2

Preference learning [Xing et al NIPS ‘02]
S-goodness [Balcan-Blum ICML ‘06]

ℓ 𝑦1,𝑦2 𝐿 = 𝜚 𝑧1𝑧2𝐿 𝑦1, 𝑦2

Kernel-target alignment [Cortes et al ICML ‘10]
Bipartite ranking, (p)AUC [Narasimhan-Agarwal ICML ‘13]

ℓ 𝑦1,𝑦2 𝑔 = 𝜚 𝑔 𝑦1 − 𝑔 𝑦2 𝑧1 − 𝑧2

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Learnin ing Obje jectiv ives in in Pairw irwise Learnin ing

Given training data 𝑦1, 𝑦2, … 𝑦𝑜
Learn

ℎ: 𝒴 × 𝒴 → 𝒵 such that ℒ ℎ ≤ ℒ ℎ∗ + 𝜗 (will define ℒ ⋅ and ℒ ⋅ shortly) Challenges:

Training data given as singletons, not pairs
Algorithmic efficiency
Generalization error bounds

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Part II: II: Batch Learning

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Part II: II: Batch Learning

Batch Learning for Unary Losses

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Trainin ing wit ith Unary Loss Functions

Notion of empirical loss

ℒ: ℋ → ℝ+

Given training data 𝑇 = 𝑦1, … , 𝑦𝑜 , natural notion

ℒ𝑇 ⋅ = 1 𝑜 ∑ℓ ⋅, 𝑦𝑗

Empirical risk minimization dictates us to find

ℎ, s.t. ℒ𝑇 ℎ ≤ inf

ℎ∈ℋ

ℒ𝑇 ℎ

Note that

ℒ ⋅ is a U-statistic

U-statistic: a notion of “training loss”

ℒ𝑇: ℋ → ℝ+ s.t. ∀ℎ ∈ ℋ, 𝔽 ℒ𝑇 ℎ = ℒ ℎ

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Generali lization bounds for Unary ry Loss Functio ions

Step 1: Bound excess risk by suprēmus excess risk

ℒ ℎ − ℒ𝑇 ℎ ≤ sup

ℎ∈ℋ

ℒ ℎ − ℒ𝑇 ℎ

Step 2: Apply McDiarmid’s inequality

ℒ𝑇 ℎ is not perturbed by changing any 𝑦𝑗 ℒ ℎ − ℒ𝑇 ℎ ≤ 𝔽 sup

ℎ∈ℋ

ℒ ℎ − ℒ𝑇 ℎ + 𝒫 1 𝑜

Step 3: Analyze the expected suprēmus excess risk

𝔽 sup

ℎ∈ℋ

ℒ ℎ − ℒ𝑇 ℎ = 𝔽 sup

ℎ∈ℋ

𝔽 ℒ

𝑇 ℎ

− ℒ𝑇 ℎ ≤ 𝔽 sup

ℎ∈ℋ

ℒ

𝑇 ℎ −

ℒ𝑇 ℎ (Jensen′s inequality)

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Analy lyzin ing th the Expected Supr uprēmus Excess Ris isk

For unary losses

ℒ𝑇 ⋅ = ∑ℓ𝑦𝑗 ⋅

Analyzing this term through symmetrization easy

1 n 𝔽 sup

ℎ∈ℋ

∑ℓ𝑦𝑗 ℎ − ℓ

𝑦𝑗 ℎ

≤ 2 𝑜 𝔽 sup

ℎ∈ℋ

∑𝜗𝑗ℓ𝑦𝑗 ℎ ≤ 2𝑀 𝑜 𝔽 sup

ℎ∈ℋ

∑𝜗𝑗ℎ 𝑦𝑗 ≈ 𝒫 1 𝑜

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𝔽 sup

ℎ∈ℋ

ℒ

𝑇 ℎ −

ℒ𝑇 ℎ

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Part II: II: Batch Learning

Batch Learning for Pairwise Loss Functions

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Trainin ing wit ith Pairw irwis ise Loss Functions

Given training data 𝑦1, 𝑦2, … 𝑦𝑜, choose a U-statistic
U-statistic should use terms like ℓ 𝑦𝑗,𝑦𝑘

ℎ (the kernel)

Population risk defined as ℒ ⋅ = 𝔽ℓ 𝑦,𝑦′

⋅ Examples:

For any index set Ω ⊂ 𝑜 × 𝑜 , define

ℒS ⋅; Ω = 1 Ω

𝑗,𝑘 ∈Ω

ℓ 𝑦𝑗,𝑦𝑘 ⋅

Choice of Ω =

𝑗, 𝑘 : 𝑗 ≠ 𝑘 maximizes data utilization

Various ways of optimizing inf

ℎ∈ℋ

ℒ𝑇 ℎ (e.g. SSG)

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Generali lization bounds for Pairw irwise Loss Functio ions

Step 1: Bound excess risk by suprēmus excess risk

ℒ ℎ − ℒ𝑇 ℎ ≤ sup

ℎ∈ℋ

ℒ ℎ − ℒ𝑇 ℎ

Step 2: Apply McDiarmid’s inequality

Check that ℒ𝑇 ℎ is not perturbed by changing any 𝑦𝑗 ℒ ℎ − ℒ𝑇 ℎ ≤ 𝔽 sup

ℎ∈ℋ

ℒ ℎ − ℒ𝑇 ℎ + 𝒫 1 𝑜

Step 3: Analyze the expected suprēmus excess risk

𝔽 sup

ℎ∈ℋ

ℒ ℎ − ℒ𝑇 ℎ = 𝔽 sup

ℎ∈ℋ

𝔽 ℒ

𝑇 ℎ

− ℒ𝑇 ℎ ≤ 𝔽 sup

ℎ∈ℋ

ℒ

𝑇 ℎ −

ℒ𝑇 ℎ (Jensen′s inequality)

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Analy lyzin ing th the Expected Supr uprēmus Excess Ris isk

For pairwise losses

ℒ𝑇 ⋅ = ∑𝑗≠𝑘 ℓ 𝑦𝑗,𝑦𝑘 ⋅

Clean symmetrization not possible due to coupling

2𝔽 sup

ℎ∈ℋ 𝑗 𝑘

ℓ

𝑦𝑗, 𝑦𝑘

ℎ − ℓ 𝑦𝑗,𝑦𝑘 ℎ

Solutions [see Clémençon et al Ann. Stat. ‘08]
Alternate representation of U-statistics
Hoeffding decomposition

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𝔽 sup

ℎ∈ℋ

ℒ

𝑇 ℎ −

ℒ𝑇 ℎ

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Part III III: Onli line Learning

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Part III III: Onli line Learning

A Whirlwind Tour of Online Learning for Unary Losses

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Model l for Onli line Learnin ing wit ith Unary Losses

Propose hypothesis ℎ𝑢−1 ∈ ℋ Receive loss ℓ𝑢 ⋅ = ℓ 𝑦𝑢,⋅ Update ℎ𝑢−1 → ℎ𝑢

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Regret

ℜ𝑈 = ∑ℓ𝑢 ℎ𝑢−1 − inf

ℎ∈ℋ ∑ℓ𝑢 ℎ

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Onli line Learnin ing Alg lgorit ithms

Generalized Infinitesimal Gradient Ascent (GIGA)

[Zinkevich ’03] ℎ𝑢 = ℎ𝑢−1 − 𝜃𝑢𝛼ℎℓ𝑢 ℎ𝑢−1

Follow the Regularized Leader (FTRL)

[Hazan et al ‘06] ℎ𝑢 = argmin

ℎ∈ℋ 𝜐=1 𝑢−1

ℓ𝜐 ℎ + 𝜏𝑢 ℎ 2

Under some conditions

ℜ𝑈 ≤ 𝒫 𝑈

Under stronger conditions

ℜ𝑈 ≤ 𝒫 log 𝑈

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Onli line to Batch Conversio ion for Unary Losses

Key insight: ℎ𝑢−1 is evaluated on an unseen point

[Cesa-Bianchi et al ‘01] 𝔽 ℓ𝑢 ℎ𝑢−1 |𝜏(𝑦1, … , 𝑦𝑢−1) = 𝔽ℓ ℎ𝑢−1, 𝑦𝑢 = ℒ ℎ𝑢−1

Set up a martingale difference sequence

𝑊

𝑢 = ℒ ℎ𝑢−1 − ℓ𝑢 ℎ𝑢−1

𝔽 𝑊

𝑢|𝜏 𝑦1, … , 𝑦𝑢−1

= 0

Azuma-Hoeffding gives us

∑ℒ ℎ𝑢−1 ≤ ∑ℓ𝑢 ℎ𝑢−1 + 𝒫 𝑈 ∑ℓ𝑢 ℎ∗ ≥ 𝑈ℒ ℎ∗ − 𝒫 𝑈

Together we get

∑ℒ ℎ𝑢−1 − 𝑈ℒ ℎ∗ ≤ ℜ𝑈 + 𝒫 𝑈

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Onli line to Batch Conversio ion for Unary Losses

Hypothesis selection
Convex loss function

ℎ =

1 𝑈 ∑ℎ𝑢

ℒ ℎ ≤ 1 𝑈 ∑ℒ ℎ𝑢 ≤ ℒ ℎ∗ + ℜ𝑈 𝑈 + 𝒫 1 𝑈

More involved for non convex losses
Better results possible [Tewari-Kakade ‘08]
Assume strongly convex loss functions

∑ℒ ℎ𝑢−1 ≤ 𝑈ℒ ℎ∗ + ℜ𝑈 + 𝒫 ℜ𝑈

For ℜ𝑈 = 𝒫 log 𝑈 , this reduces to

ℒ ℎ ≤ 1 𝑈 ∑ℒ ℎ𝑢 ≤ ℒ ℎ∗ + 𝒫 log 𝑈 𝑈

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Part III III: Onli line Learning

Online Learning for Pairwise Loss Functions

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Model l for Onli line Learnin ing wit ith Pairw irwise Losses

Propose hypothesis ℎ𝑢−1 ∈ ℋ Receive loss ℓ𝑢 ⋅ = ? Update ℎ𝑢−1 → ℎ𝑢

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Regret

ℜ𝑈 = ?

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Defin inin ing In Instantaneous Loss and Regret

At time 𝑢, we receive point 𝑦𝑢
Natural definition of instantaneous loss:

All the pairwise interactions 𝑦𝑢 has with previous points ℓ𝑢 ⋅ =

𝜐=1 𝑢−1

ℓ 𝑦𝑢,𝑦𝜐 ⋅

Corresponding notion of regret

ℜ𝑈 = ∑ℓ𝑢 ℎ𝑢−1 − inf

ℎ∈ℋ ∑ℓ𝑢 ℎ

Note that this notion of instantaneous loss satisfies

∀ℎ ∈ ℋ, ∑ℓ𝑢 ℎ =

𝑗<𝑘

ℓ 𝑦𝑗,𝑦𝑘 ℎ = 1 2 ℒ𝑇 ℎ

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Onli line Learnin ing Alg lgorit ithm wit ith Pairw irwise Losses

For regularity, we use a normalized loss

ℓ𝑢 ⋅ = 1 𝑢 − 1

𝜐=1 𝑢−1

ℓ 𝑦𝑢,𝑦_𝜐 ⋅

Note that ℓ𝑢 ⋅ is convex, bounded and Lipchitz if ℓ is so
Turns out GIGA works just fine

ℎ𝑢 = ℎ𝑢−1 − 𝜃𝑢𝛼ℎℓ𝑢 ℎ𝑢−1

Guarantees similar regret bounds

ℜ𝑈 ≤ 𝒫 𝑈

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Onli line Learnin ing Alg lgorit ithm wit ith Pairw irwise Losses

Implementing GIGA requires storing previous history

𝛼ℎℓ𝑢 ⋅ = 1 𝑢 − 1

𝜐=1 𝑢−1

𝛼ℎℓ 𝑦𝑢,𝑦𝜐 ⋅

To reduce memory usage, keep a snapshot of history
Limited memory buffer 𝐶 = □1, □2, … , □𝑡
Modified instantaneous loss

ℓ𝑢

buf ⋅ = 1

𝑡

𝑦∈𝐶𝑢−1

ℓ 𝑦𝑢,𝑦 ⋅

Responsibilities: at each time step 𝑢
Update hypothesis ℎ𝑢−1 → ℎ𝑢 (same as GIGA but with ℓ𝑢

buf ⋅ )

Update buffer UPDATE 𝐶𝑢−1, 𝑦𝑢 → 𝐶𝑢

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Buffer Update Alg lgorithm

Online sampling algorithm for i.i.d. samples

[K. et al ‘13]

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RS-x: Reservoir sampling with replacement

𝑨1 𝑨𝑢 𝑨7 𝑨1 𝑨2 𝑨2 𝑨3 𝑨3 𝑨6 𝑨4 𝑨5 𝑨𝑢

∼ 𝑪 𝟐 𝒖

𝑨𝑢 𝑨𝑢

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Regret Analy lysis is for GIG IGA wit ith RS-x

RS-x gives the following guarantee

At any fixed time 𝑢, the buffer 𝐶 contains 𝑡 i.i.d. samples from the previous history 𝐼𝑢 = 𝑦1, … , 𝑦𝑢−1

Use this to prove a Regret Conversion Bound
Basic idea
Prove a finite buffer regret bound

1 T ∑ℓ𝑢

buf ℎ𝑢−1 ≤ inf ℎ∈ℋ

1 𝑈 ∑ℓ𝑢

buf ℎ + 𝒫

1 𝑈

Use uniform convergence style bounds to show

ℓ𝑢 ℎ𝑢−1 ≈ ℓ𝑢

buf ℎ𝑢−1 ±

𝒫 1 𝑡

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Regret Analy lysis is for GIG IGA wit ith RS RS-x: Step 1

Finite Buffer Regret

The modified algo. uses ℓ𝑢

buf ⋅ to update hypothesis

ℓ𝑢

buf ⋅ is also convex, bounded and Lipchitz given 𝐶

Standard GIGA analysis gives us

1 T ∑ℓ𝑢

buf ℎ𝑢−1 ≤ inf ℎ∈ℋ

1 𝑈 ∑ℓ𝑢

buf ℎ + 𝒫 ℜ𝑈 buf

𝑈 , where ℜ𝑈

buf = 𝒫

𝑈

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Regret Analy lysis is for GIG IGA wit ith RS RS-x: Step 2

Uniform convergence

Think of 𝐼𝑢 as population and 𝐶 as i.i.d. sample of size 𝑡
Define 𝑕𝑦 ⋅ = ℓ 𝑦𝑢,𝑦 ⋅ and set unif. dist. over 𝐼𝑢
Population risk analysis

𝒣 ⋅ = 𝔽𝑕𝑦 ⋅ = 1 𝑢 − 1

𝜐=1 𝑢−1

ℓ 𝑦𝑢,𝑦𝜐 ⋅ = ℓ𝑢 ⋅

Empirical risk analysis

𝒣 ⋅ = 1 𝑡

𝑦∈𝐶𝑢−1

𝑕 𝑦 ⋅ = 1 𝑡

𝑦∈𝐶𝑢−1

ℓ 𝑦𝑢,𝑦 ⋅ = ℓ𝑢

buf ⋅

Finish off using 𝒣 ⋅ −

𝒣 ⋅

∞ ≤

𝒫

1 𝑡

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Regret Analy lysis is for GIG IGA wit ith RS-x: Wrappin ing up

Convert finite buffer regret to true regret
Three results:

∀𝑢, ℓ𝑢 ℎ𝑢−1 ≤ ℓ𝑢

buf ℎ𝑢−1 +

𝒫 1 𝑡 ∀ℎ, ∀𝑢, ℓ𝑢

buf ℎ ≤ ℓ𝑢 ℎ +

𝒫 1 𝑡 ∀ℎ, 1 𝑈 ∑ℓ𝑢

buf ℎ𝑢−1 ≤ 1

𝑈 ∑ℓ𝑢

buf ℎ + ℜ𝑈 buf

𝑈

Combine to get

1 T ∑ℓ𝑢 ℎ𝑢−1 ≤ inf

ℎ∈ℋ

1 𝑈 ∑ℓ𝑢 ℎ + 𝒫 1 𝑡 + ℜ𝑈

buf

𝑈 i.e. ℜ𝑈 ≤ ℜ𝑈

buf +

𝒫

𝑈 𝑡 =

𝒫

𝑈 𝑡

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Regret Analy lysis is for GIG IGA wit ith RS-x

Better results possible for strongly convex losses
For any 𝜗 > 0, we can show

1 T ∑ℓ𝑢 ℎ𝑢−1 ≤ 1 + 𝜗 inf

ℎ∈ℋ

∑ℓ𝑢 ℎ 𝑈 + ℜ𝑈 𝑈 + 𝒫 1 𝜗𝑡

For realizable cases (i.e. ℒ ℎ∗ = 0), we can also show

1 T ∑ℓ𝑢 ℎ𝑢−1 ≤ inf

ℎ∈ℋ

1 𝑈 ∑ℓ𝑢 ℎ + ℜ𝑈 𝑈 + 𝒫 ℜ𝑈 𝑡

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Onli line to Batch Conversio ion for Pair irwise Losses

Recall that in unary case, we had an MDS

𝑊

𝑢 = ℒ ℎ𝑢−1 − ℓ𝑢 ℎ𝑢−1

Recall, in pairwise case, we have

ℒ ⋅ = 𝔽ℓ 𝑦,𝑦′ ⋅ ℓ𝑢 ⋅ = 1 𝑢 − 1

𝜐=1 𝑢−1

ℓ 𝑦𝑢,𝑦𝜐 ⋅

No longer an MDS since 𝑊

𝑢 and 𝑊 𝜐, 𝜐 < 𝑢 are coupled

𝔽 𝑊

𝑢|𝜏 𝐼𝑢

= ℒ ℎ𝑢−1 − 𝔽 ℓ𝑢 ℎ𝑢−1 |𝜏 𝐼𝑢 ≠ 0

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Onli line to Batch Conversio ion for Pair irwise Losses

Solution:

Martingale creation: let

ℓ𝑢 ⋅ = 𝔽 ℓ𝑢 ⋅ |𝜏 𝐼𝑢 𝑊

𝑢 = ℒ ℎ𝑢−1 −

ℓ𝑢 ℎ𝑢−1 + ℓ𝑢 ℎ𝑢−1 − ℓ𝑢 ℎ𝑢−1 𝑊

𝑢 = 𝑄𝑢 + 𝑅𝑢

Sequence 𝑅𝑢 is an MDS by construction: A.H. bounds
Bounding 𝑄𝑢 using uniform convergence
Be careful during symmetrization step
End Result

1 𝑈 ∑ℒ ℎ𝑢−1 ≤ ℒ ℎ∗ + ℜ𝑈 𝑈 + 𝒫 1 𝑈

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Faster Rates for Str trongly ly Convex Losses

Have to use fast rate results to bound both 𝑄

𝑢 and 𝑅𝑢

Fast rates for 𝑄𝑢

For strongly unary convex loss functions ℓ𝑦 ⋅ , we have ℒ ℎ − ℒ ℎ∗ ≤ 1 + 𝜗 ℒ𝑇 ℎ − ℒ𝑇 ℎ∗ + 𝒫 1 𝜗𝑜

Fast rates for 𝑅𝑢

Use Bernstein inequality for martingales

End result

1 𝑈 ∑ℒ ℎ𝑢−1 ≤ ℒ ℎ∗ + ℜ𝑈 𝑈 + 𝒫 ℜ𝑈 𝑈

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Hid idden Constants

All our analyses involved Rademacher averages
Even for regret analysis and bounding 𝑄𝑢 for slow/fast rates
Get dimension independent bounds for regularized classes
Weak dependence on dimensionality for sparse formulations
Earlier work [Wang et al ‘12] used covering number methods
If constants not imp. then can try analyzing 𝑊

𝑢 directly

Use covering number arguments to get linear dep. on 𝑒

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Some In Interesting Proje jects

Regret bounds require 𝑡 = 𝜕 log 𝑈
Is this necessary: regret lower bound
Learning higher order tensors
Scalability issues
RS-x is a data oblivious sampling algorithm
Can throw away useful points by chance
Data aware sampling methods + corresponding regret bounds

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That’s all!

Get slides from the following URL http://research.microsoft.com/en-us/people/t-purkar/

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References

Balcan, Maria-Florina and Blum, Avrim. On a Theory of Learning

with Similarity Functions. In ICML, pp. 73-80, 2006.

Cesa-Bianchi, Nicolo, Conconi, Alex, and Gentile, Claudio. On the

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