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The Value of Reward Lookahead in Reinforcement Learning
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Uses competitive analysis to quantify the value of partial future-reward lookahead in tabular episodic MDPs, deriving exact worst-case competitive ratios (via long-shot rewards) and tight bounds over worst-case environments.
Problem
How much value do RL agents lose relative to agents that observe future rewards L steps ahead before acting, measured as a worst-case competitive ratio over reward distributions, expectations, and environments.
Approach
Measure the competitive ratio between no-lookahead agents (who know only reward expectations) and L-step lookahead agents; show worst-case rewards are long-shot distributions, yielding exact ratios expressed via occupancy measures that coincide with coverability coefficients from offline and reward-free RL.
Key results
- Worst-case competitive ratio versus full lookahead satisfies CR^H(P) equal to the inverse coverability coefficient, with lower bound max{1/(SAH), 1/A^H} for all environments.
- For one-step lookahead, CR^1 = Theta(1/(HA)) even for the worst-case environment, independent of state-space size.
- For chain MDPs the prophet-type structure gives CR of at least 1/((A-1)H) versus full lookahead, and CR^1(P) is at least (1-1/e)/(AH) for chains.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| provide L-step future reward lookahead instead of no lookahead | tabular episodic reinforcement learning agents · Worst-case environments | competitive ratio: CR^L\ge \max\brc*{\frac{1}{SAH},\frac{1}{(H-L+1)A^L}} | Hurt |
| increase lookahead from one step to full lookahead | tabular episodic reinforcement learning agents · Worst-case environment | competitive ratio: CR^1=\Theta\br*{\frac{1}{HA}} | Hurt |
| add reward lookahead in a disguised contextual bandit | disguised contextual bandit · Disguised contextual bandit | competitive ratio: CR^L(P)=\frac{1}{A} | No clear effect |
| add a delaying action at the tree root | delayed trees · Worst-case environments | competitive ratio: CR^L(P,r)\!=\!\Theta\br*{\max\brc*{\frac{1}{(H-L)A^L},\frac{1}{SAH}}} | Hurt |
| use dense rewards with within-state reward ratio bounded by C | one-step lookahead agents · Dense rewards | competitive ratio: CR^1(P,r)\geq \frac{1}{AC} | Helped |
| use near-uniform transitions with probabilities in the stated interval | full-lookahead comparison · Near-uniform transitions | competitive ratio: CR^H(P) \ge \frac{1}{C AH} | Helped |
| make all grid reward expectations unit-valued | grid MDP · Grid MDP | competitive ratio: CR^L(P)=\Theta(\frac{1}{L}) | Mixed |
Surprising findings
- The competitive ratio for worst-case reward expectations relates to the coverability coefficient from offline and reward-free RL, despite the lookahead setting having no learning component.
- One-step lookahead can have exponentially small competitive ratio when transition lookahead is considered instead of reward lookahead (claimed in conclusions).
Limitations
- Results are purely theoretical in tabular episodic MDPs with known dynamics and reward distributions.
- No practical algorithm is given for achieving lookahead value; planning with general reward distributions is hard.
Experimental setup
Metrics: competitive ratio
Builds on: prophet inequality coverability coefficient model predictive control
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