Efficient Online Portfolio with Logarithmic Regret Haipeng Luo - - PowerPoint PPT Presentation

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Efficient Online Portfolio with Logarithmic Regret Haipeng Luo - - PowerPoint PPT Presentation

Jun 03, 2023 134 likes β€’486 views

Efficient Online Portfolio with Logarithmic Regret Haipeng Luo (USC) Chen-Yu Wei (USC) Kai Zheng (Peking University) Online Portfolio Wealth Online Portfolio 0.5Wealth 0.3Wealth Wealth 0.2Wealth Online Portfolio

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SLIDE 1

Efficient Online Portfolio with Logarithmic Regret

Haipeng Luo (USC) Chen-Yu Wei (USC) Kai Zheng (Peking University)

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SLIDE 2

Online Portfolio

Wealth𝑒

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SLIDE 3

Online Portfolio

Wealth𝑒 0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒

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SLIDE 4

Online Portfolio

Wealth𝑒 0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒

Γ— 1.4 Γ— 1.0 Γ— 0.5

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SLIDE 5

Online Portfolio

Wealth𝑒 0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒 0.7Wealth𝑒 0.3Wealth𝑒 0.1Wealth𝑒

Γ— 1.4 Γ— 1.0 Γ— 0.5

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SLIDE 6

Online Portfolio

Wealth𝑒+1 = 0.7 + 0.3 + 0.1 Wealth𝑒 = 1.1Wealth𝑒 Wealth𝑒 0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒 0.7Wealth𝑒 0.3Wealth𝑒 0.1Wealth𝑒

Γ— 1.4 Γ— 1.0 Γ— 0.5

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SLIDE 7

Online Portfolio

Wealth𝑒

𝑦𝑒 (decision)

0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒 0.7Wealth𝑒 0.3Wealth𝑒 0.1Wealth𝑒

Γ— 1.4 Γ— 1.0 Γ— 0.5

Wealth𝑒+1 = 0.7 + 0.3 + 0.1 Wealth𝑒 = 1.1Wealth𝑒

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SLIDE 8

Online Portfolio

Wealth𝑒+1 = 0.7 + 0.3 + 0.1 Wealth𝑒 = 1.1Wealth𝑒 Wealth𝑒

𝑦𝑒 (decision) 𝑠

𝑒 (price relative)

0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒 0.7Wealth𝑒 0.3Wealth𝑒 0.1Wealth𝑒

Γ— 1.4 Γ— 1.0 Γ— 0.5

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SLIDE 9

Online Portfolio

Wealth𝑒+1 = 0.7 + 0.3 + 0.1 Wealth𝑒 = 1.1Wealth𝑒 = 𝑦𝑒, 𝑠𝑒 Wealth𝑒 Wealth𝑒

𝑦𝑒 (decision) 𝑠

𝑒 (price relative)

0.5Wealth𝑒 0.3Wealth𝑒 0.2Wealth𝑒 0.7Wealth𝑒 0.3Wealth𝑒 0.1Wealth𝑒

Γ— 1.4 Γ— 1.0 Γ— 0.5

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SLIDE 10

Online Portfolio

𝑋

1

π‘ˆ periods

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SLIDE 11

Online Portfolio

𝑋

1

𝑋

2

= 𝑦1, 𝑠

1 𝑋 1

π‘ˆ periods

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SLIDE 12

Online Portfolio

𝑋

1

𝑋

2

= 𝑦1, 𝑠

1 𝑋 1

𝑋

3

= 𝑦2, 𝑠

2 𝑋 2

π‘ˆ periods

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SLIDE 13

Online Portfolio

𝑋

1

𝑋

2

= 𝑦1, 𝑠

1 𝑋 1

𝑋

3

= 𝑦2, 𝑠

2 𝑋 2

π‘ˆ periods

π‘‹π‘ˆ+1 𝑋

1

=

𝑒=1 π‘ˆ

𝑦𝑒, 𝑠

𝑒

Final wealth Initial wealth

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SLIDE 14

Online Portfolio

Gain:

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SLIDE 15

Online Portfolio

Gain: Benchmark:

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SLIDE 16

Online Portfolio

Gain: Benchmark: Minimize (Regret)

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SLIDE 17

Online Portfolio

Gain:

Online Convex Optimization [Zinkevich’03]

Benchmark: Minimize (Regret)

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SLIDE 18

Online Portfolio

Gain:

Online Convex Optimization [Zinkevich’03]

Benchmark: Minimize

Maximum Relative Ratio

𝛼ℓ𝑒 𝑦

∞ β‰Ύ 𝐻 β‰œ max 𝑗,π‘˜

𝑠

𝑒,𝑗

𝑠

𝑒,π‘˜

(Regret)

But with possibly unbounded gradient

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SLIDE 19

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

𝑂: number of stocks π‘ˆ: number of rounds

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SLIDE 20

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 21

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 22

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4 ONS (Hazan et al. 2007) 𝐻𝑂 log π‘ˆ 𝑂3.5

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 23

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4 ONS (Hazan et al. 2007) 𝐻𝑂 log π‘ˆ 𝑂3.5

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 24

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4 ONS (Hazan et al. 2007) 𝐻𝑂 log π‘ˆ 𝑂3.5 Soft-Bayes (Orseau et al. 2017) π‘ˆπ‘‚ 𝑂

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 25

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4 ONS (Hazan et al. 2007) 𝐻𝑂 log π‘ˆ 𝑂3.5 Soft-Bayes (Orseau et al. 2017) π‘ˆπ‘‚ 𝑂

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 26

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4 ONS (Hazan et al. 2007) 𝐻𝑂 log π‘ˆ 𝑂3.5 Soft-Bayes (Orseau et al. 2017) π‘ˆπ‘‚ 𝑂 ? β‰ˆ 𝑂 log π‘ˆ β‰ˆ 𝑂

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 27

Previous Results and Our Results

  • Lower bound: Ξ© 𝑂 log π‘ˆ

Algorithm Regret Time (/round) Universal Portfolio

(Cover 1991, Kalai et al. 2002)

𝑂 log π‘ˆ π‘ˆ14𝑂4 ONS (Hazan et al. 2007) 𝐻𝑂 log π‘ˆ 𝑂3.5 Soft-Bayes (Orseau et al. 2017) π‘ˆπ‘‚ 𝑂 ? β‰ˆ 𝑂 log π‘ˆ β‰ˆ 𝑂 BarrONS (this work) 𝑂2 log π‘ˆ 4 π‘ˆπ‘‚2.5

  • Upper bounds:

𝑂: number of stocks π‘ˆ: number of rounds 𝐻: maximum relative ratio

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SLIDE 28

Key Components of Our Algorithm

bad bad suddenly good But player puts little weight on it Main Challenge:

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SLIDE 29

Key Components of Our Algorithm

Barrons (Barrier-Regularized-ONS) compared to ONS:

bad bad suddenly good But player puts little weight on it Main Challenge:

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SLIDE 30

Key Components of Our Algorithm

Barrons (Barrier-Regularized-ONS) compared to ONS: 1. Additional regularizer (to avoid too extreme distribution over stocks)

bad bad suddenly good But player puts little weight on it Main Challenge:

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SLIDE 31

Key Components of Our Algorithm

Barrons (Barrier-Regularized-ONS) compared to ONS: 1. Additional regularizer (to avoid too extreme distribution over stocks) 2. Increase the learning rate for worse stocks (faster recovery)

bad bad suddenly good But player puts little weight on it Main Challenge:

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SLIDE 32

Key Components of Our Algorithm

Barrons (Barrier-Regularized-ONS) compared to ONS: 1. Additional regularizer (to avoid too extreme distribution over stocks) 2. Increase the learning rate for worse stocks (faster recovery) 3. Restarting (adapting to maximum relative ratio)

bad bad suddenly good But player puts little weight on it Main Challenge:

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SLIDE 33

Key Components of Our Algorithm

Barrons (Barrier-Regularized-ONS) compared to ONS: 1. Additional regularizer (to avoid too extreme distribution over stocks) 2. Increase the learning rate for worse stocks (faster recovery) 3. Restarting (adapting to maximum relative ratio)

bad bad suddenly good But player puts little weight on it Main Challenge:

Poster #157