Efficient Online Portfolio with Logarithmic Regret

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

Online Portfolio

Wealth𝑡

Online Portfolio

Wealth𝑡

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

Online Portfolio

Wealth𝑡

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

× 1.4

× 1.0

× 0.5

Online Portfolio

Wealth𝑡

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

0.7Wealth𝑡

0.3Wealth𝑡

0.1Wealth𝑡

× 1.4

× 1.0

× 0.5

Online Portfolio

Wealth𝑡+1= 0.7 + 0.3 + 0.1 Wealth𝑡= 1.1Wealth𝑡

Wealth𝑡

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

0.7Wealth𝑡

0.3Wealth𝑡

0.1Wealth𝑡

× 1.4

× 1.0

× 0.5

Online Portfolio

Wealth𝑡

𝑥𝑡 (decision)

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

0.7Wealth𝑡

0.3Wealth𝑡

0.1Wealth𝑡

× 1.4

× 1.0

× 0.5

Wealth𝑡+1= 0.7 + 0.3 + 0.1 Wealth𝑡= 1.1Wealth𝑡

Online Portfolio

Wealth𝑡+1= 0.7 + 0.3 + 0.1 Wealth𝑡= 1.1Wealth𝑡

Wealth𝑡

𝑥𝑡 (decision) 𝑟𝑡 (price relative)

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

0.7Wealth𝑡

0.3Wealth𝑡

0.1Wealth𝑡

× 1.4

× 1.0

× 0.5

Online Portfolio

Wealth𝑡+1= 0.7 + 0.3 + 0.1 Wealth𝑡= 1.1Wealth𝑡= 𝑥𝑡, 𝑟𝑡 Wealth𝑡

Wealth𝑡

𝑥𝑡 (decision) 𝑟𝑡 (price relative)

0.5Wealth𝑡

0.3Wealth𝑡

0.2Wealth𝑡

0.7Wealth𝑡

0.3Wealth𝑡

0.1Wealth𝑡

× 1.4

× 1.0

× 0.5

Online Portfolio

𝑊1

𝑇 periods

Online Portfolio

𝑊1 𝑊2= 𝑥1, 𝑟1 𝑊1

𝑇 periods

Online Portfolio

𝑊1 𝑊2= 𝑥1, 𝑟1 𝑊1

𝑊3= 𝑥2, 𝑟2 𝑊2

𝑇 periods

Online Portfolio

𝑊1 𝑊2= 𝑥1, 𝑟1 𝑊1

𝑊3= 𝑥2, 𝑟2 𝑊2

𝑇 periods

𝑊𝑇+1𝑊1=

𝑡=1

𝑇

𝑥𝑡 , 𝑟𝑡Final wealth

Initial wealth

Online Portfolio

Gain:

Online Portfolio

Gain:

Benchmark:

Online Portfolio

Gain:

Benchmark:

Minimize (Regret)

Online Portfolio

Gain:

Online Convex Optimization [Zinkevich’03]Benchmark:

Minimize (Regret)

Online Portfolio

Gain:

Online Convex Optimization [Zinkevich’03]Benchmark:

Minimize

Maximum Relative Ratio

𝛻ℓ𝑡 𝑥 ∞ ≾ 𝐺 ≜ max𝑖,𝑗

𝑟𝑡,𝑖𝑟𝑡,𝑗

(Regret)

But with possibly unbounded gradient

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇 𝑁: number of stocks

𝑇: number of rounds

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

ONS (Hazan et al. 2007) 𝐺𝑁 log 𝑇 𝑁3.5

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

ONS (Hazan et al. 2007) 𝐺𝑁 log 𝑇 𝑁3.5

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

ONS (Hazan et al. 2007) 𝐺𝑁 log 𝑇 𝑁3.5

Soft-Bayes (Orseau et al. 2017) 𝑇𝑁 𝑁

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

ONS (Hazan et al. 2007) 𝐺𝑁 log 𝑇 𝑁3.5

Soft-Bayes (Orseau et al. 2017) 𝑇𝑁 𝑁

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

ONS (Hazan et al. 2007) 𝐺𝑁 log 𝑇 𝑁3.5

Soft-Bayes (Orseau et al. 2017) 𝑇𝑁 𝑁

? ≈ 𝑁 log 𝑇 ≈ 𝑁

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Previous Results and Our Results

• Lower bound: Ω 𝑁 log 𝑇

Algorithm Regret Time (/round)

Universal Portfolio (Cover 1991, Kalai et al. 2002)

𝑁 log 𝑇 𝑇14𝑁4

ONS (Hazan et al. 2007) 𝐺𝑁 log 𝑇 𝑁3.5

Soft-Bayes (Orseau et al. 2017) 𝑇𝑁 𝑁

? ≈ 𝑁 log 𝑇 ≈ 𝑁

BarrONS (this work) 𝑁2 log 𝑇 4 𝑇𝑁2.5

• Upper bounds:

𝑁: number of stocks

𝑇: number of rounds

𝐺: maximum relative ratio

Key Components of Our Algorithm

bad bad suddenly good

But player puts little weight on it

Main Challenge:

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:

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:

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:

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:

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