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Contextual Online Pricing with (Biased) Offline Data
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Studies contextual online pricing with biased offline data, proposing OFU-based CO3/GCO3 algorithms with instance-dependent (delta^2) and minimax regret bounds, plus a robust RCO3 variant for unknown bias bound.
Problem
Contextual online pricing when historical (offline) pricing data come from a biased, shifted demand model; the firm must learn to price with regret that improves over purely online methods when the bias is small.
Approach
Uses a three-ellipsoid confidence set (online Gram ellipsoid, combined online-offline ellipsoid, and an offline Euclidean ball bounded by the bias V) inside an OFU policy, plus an offline testing phase that decides whether to deploy the empirical offline price policy; introduces the instance-dependent distance delta^2 between the offline OLS price policy and the true optimal price.
Key results
- Upper bound for CO3 in the scalar elasticity case is O~(d1 sqrt(T) ∧ (V^2 T + d1 T/(lambda_min + (N∧T) delta^2))), matched by a lower bound up to a factor of d1.
- GCO3 achieves worst-case O~((d1+d2) sqrt(T) ∧ (V^2 T + (d1+d2)T/lambda_min)), matching a lower bound up to a linear factor in dimension.
- In simulations with T=1000 and 20 trials, CO3/GCO3 outperform UCB, TS and offline baselines under a tight bias bound and match UCB under a loose bound.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CO3 with biased offline data and a tight bias bound instead of the baselines | Scalar price elasticity pricing · CB-OPOD, scalar price elasticity | cumulative regret | Helped |
| Use GCO3 with biased offline data and a tight bias bound instead of the baselines | General price elasticity pricing · CB-OPOD, general price elasticity | cumulative regret | Helped |
| Use UCB-Offline, which combines offline and online data in a single confidence ellipsoid, instead of pure online UCB | Online pricing with biased offline data · CB-OPOD | cumulative regret | Hurt |
| Use TS-Offline with a prior fitted to offline data instead of pure online TS | Online pricing with biased offline data · CB-OPOD | cumulative regret | Hurt |
| Use CO3/GCO3 with a loose bias bound instead of pure online UCB | Contextual online pricing · CB-OPOD, loose bias bound | cumulative regret | No clear effect |
| Use RCO3 with offline data when the exact bias is small instead of pure online UCB | General contextual online pricing · CB-OPOD, small exact bias | cumulative regret | Helped |
| Use RCO3 with offline data when the exact bias is large instead of pure online UCB | General contextual online pricing · CB-OPOD, large exact bias | cumulative regret | No clear effect |
| Increase offline data dispersion when the bias bound is small | CO3 · CB-OPOD, scalar price elasticity | regret | Helped |
| Increase the bias bound when using offline data | CO3 · CB-OPOD, scalar price elasticity | regret | Hurt |
Surprising findings
- Offline-data-based baselines (UCB-Offline, TS-Offline) accumulate regret faster than pure-online baselines under biased offline data.
- Without bias information any policy suffers Omega(sqrt(T)) worst-case regret, so offline data alone cannot beat online learning in the worst case.
Limitations
- Lower bound in general case is only tight up to a linear factor in dimension d1+d2.
- Scalar-elasticity results rely on the assumption lambda_min(Sigma_xx) >= cN and the instance-dependent delta^2 is specific to the scalar case.
- Experiments are small-scale synthetic simulations with T=1000 and 20 trials.
Experimental setup
Datasets: synthetic offline pricing data
Benchmarks: synthetic linear demand pricing simulations
Metrics: cumulative regret
Baselines: UCB UCB-Offline TS TS-Offline
Builds on: OPOD (online pricing with offline data) C-OPOD UCB for linear bandits
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