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Hindsight Credit Assignment

Anna Harutyunyan, Will Dabney, Thomas Mesnard, Mohammad Azar, Bilal Piot, Nicolas Heess, Hado van Hasselt, Greg Wayne, Satinder Singh, Doina Precup, Remi Munos

arXiv:1912.02503v1cs.LGstat.ML

TL;DR

Credit assignment in reinforcement learning is challenged by trajectory variance, partial observability, temporal relevance heuristics, and limited counterfactual use of data. The paper conditions on future outcomes to estimate past actions’ relevance and rewrites value functions through this hindsight lens. It reports that the resulting algorithms address highlighted challenges on illustrative diagnostic tasks.

  • Problem

    Credit assignment must determine how actions affect future return, but trajectory randomness, partial observability, time-based relevance heuristics, and action-specific data use complicate that task.

    Source: §1 · p. 1

  • Method

    The paper uses hindsight conditioning to estimate past actions’ relevance to future outcomes, rewrites value functions this way, and derives policy-gradient algorithms from the estimators.

    Source: §1 · pp. 1–2, §3.1 · p. 3

  • Results

    The authors report that their algorithms address highlighted credit-assignment challenges on diagnostic tasks, including delayed-effect settings where naive bootstrapping is inadequate and Monte Carlo degrades under intermediate reward noise.

    Source: §1 · p. 2, §5 · p. 8

  • Takeaways & Limitations

    Hindsight distributions are discriminative rather than generative, so they need not model the full state distribution to estimate action relevance.

    Source: §4.1 · p. 6

    The return-conditioned identity assumes positive hindsight probability for the action under every possible random return.

    Source: §3.2 · p. 4

Abstract

from arXiv · show

We consider the problem of efficient credit assignment in reinforcement learning. In order to efficiently and meaningfully utilize new data, we propose to explicitly assign credit to past decisions based on the likelihood of them having led to the observed outcome. This approach uses new information in hindsight, rather than employing foresight. Somewhat surprisingly, we show that value functions can be rewritten through this lens, yielding a new family of algorithms. We study the properties of these algorithms, and empirically show that they successfully address important credit assignment challenges, through a set of illustrative tasks.

1 Introduction

The introduction frames credit assignment as deciding how actions affect future return, a question complicated by trajectory randomness, partial observability, temporal heuristics, and limited counterfactual use. It proposes hindsight-based estimators of past decisions’ relevance to outcomes, rewriting value functions and motivating new algorithms.

1 Introduction

  • Monte Carlo return estimates are inefficient when trajectories contain substantial randomness.They average future discounted rewards from taking an action in a state.Source: §1 · p. 1
  • TD bootstrapping can be biased or fail to converge when partial observability or function approximation violates the Markov assumption.The passage cites for this risk.Source: §1 · p. 1
  • TD(λ) uses temporal distance as a relevance heuristic, assigning more credit or blame to more recent actions, though that proxy can be improved by learning.Source: §1 · p. 1
  • Action-value estimates use only trajectories containing the evaluated action, rather than reusing a trajectory to update all relevant actions.The introduction identifies this as the absence of counterfactual credit assignment.Source: §1 · p. 1
  • Hindsight conditioning estimates how relevant past actions were to observed outcomes, yielding rewritten value functions, new estimators, and policy-gradient algorithms.The authors report diagnostic tasks designed to test these algorithms on highlighted credit-assignment challenges.Source: §1 · pp. 1–2

2 Background and Notation

This section defines the MDP and policy notation used to describe expected returns, advantages, and policy-gradient updates. It also notes that practical algorithms estimate these quantities with truncated returns and may use a learned value baseline.

2 Background and Notation

  • An MDP specifies state and action spaces, transition and reward functions, and a discount factor; a policy gives action probabilities by state.Source: §2 · p. 2
  • The advantage Aπ(x, a) measures the benefit of action a over policy π, and policy-gradient methods update π along the value-function gradient.Source: §2 · p. 2
  • Practical methods such as REINFORCE approximate Qπ or Aπ with truncated returns, sometimes combining them with a bootstrapped value estimate used as a baseline.Source: §2 · p. 2

3 Conditioning on the Future

Hindsight credit assignment conditions on future outcomes to estimate how relevant past actions were, rather than predicting outcomes from the current action. Rewriting value functions with these distributions supports state- and return-conditioned estimators and policy gradients that can reuse trajectories across actions.

3 Conditioning on the Future

  • Hindsight distributions reverse the credit-assignment question to ask how relevant past decisions were to an outcome, and importance weighting can make rare outcomes more useful for learning.The paper relates this approach to change of measure and importance sampling for rare events.Source: §3 · p. 3

3.1 Conditioning on Future States

  • State-conditioned hindsight weights each future reward by how relevant the initial action was to reaching that state, allowing a trajectory sampled under the policy to update values for all actions.When the relevance ratio is 1, the reward is excluded from that action’s advantage, which may reduce variance from inconsequential noise outside the agent’s control.Source: §3.1 · p. 4
  • A time-independent state-conditioned distribution can replace conditioning on the exact timestep, with β representing per-step survival probability.The passage says β can be the discount γ or a termination probability in an undiscounted problem.Source: §3.1 · p. 4

3.2 Conditioning on Future Returns

  • Return-conditioned hindsight measures action relevance to an observed return, using a scalar outcome instead of potentially high-dimensional future states, but may use less structure in the dynamics.The return-conditioned result assumes the hindsight probability is positive for every possible return.Source: §3.2 · p. 4

3.3 Policy Gradients

  • State-HCA policy gradients are written for all actions, need no baseline, and can use returns sampled from one starting action to estimate values for all actions.The passage contrasts this with the usual all-actions algorithm, which uses a critic.Source: §3.3 · p. 5

4 Algorithms

The algorithms estimate hindsight distributions from trajectories and use them in policy updates. The paper also describes a return-conditioned variant that can omit the value function, and argues that learning action-focused discriminative distributions may be easier than learning full generative models.

4 Algorithms

  • The algorithms sample the new policy-gradient expectations, with state-conditional and return-conditional variants training hindsight distributions to predict sampled actions.The state variant also trains a value function and immediate-reward estimate; the return variant trains on complete trajectories and updates with a return-conditioned advantage.Source: §4 · pp. 5–6
  • Return-conditioned HCA can omit the value function when its hindsight distribution is estimated well, making complete rollouts feasible without variance issues.Its baseline is proportional to the return and is stated not to introduce policy-gradient bias.Source: §4 · p. 6

4.1 Learning Hindsight Distributions

  • The proposed hindsight distributions are discriminative action models, avoiding the need to model the full state distribution required by generative approaches.The passage notes that the action space is usually much smaller than the state space, and that hindsight-distribution learning is supervised learning.Source: §4.1 · p. 6

5 Experiments

The diagnostic experiments examine shortcut choice, delayed consequences under partial observability, and noisy ambiguous rewards. HCA adjusts more quickly toward shortcuts, handles delayed effects and irrelevant reward noise, and remains useful in the ambiguous bandit when state identity is hidden.

Shortcut

  • HCA more quickly adjusts the policy toward the shortcut action, with counterfactual updates increasingly effective when the long path is taken more often.The task also shows that temporal distance can be a poor relevance heuristic even in a fully observable MDP.Source: Figure 3 · p. 7, §5 · p. 8

Delayed Effect

  • In the delayed-effect task, HCA carries relevant information through partial observability and reduces variance from irrelevant reward noise, unlike naive bootstrapping or noise-sensitive Monte Carlo.Return-conditional HCA performs similarly to the baseline because it must model the noise distribution.Source: Figure 4 · p. 7, §5 · p. 8

Ambiguous Bandit

  • In the ambiguous bandit, both HCA methods improve on policy gradient when state identity is observed; with identity hidden, return HCA still improves while state HCA is handicapped.The task tests noisy, overlapping reward distributions and crossover between the action-associated states.Source: Figure 5 · p. 7, §5 · p. 8

6 Related Work

Related work connects hindsight credit assignment to goal-conditioned learning, backward credit propagation, variance reduction, and inverse models. The paper distinguishes its use of outcome conditioning for credit assignment from methods that use conditioning for other purposes.

Related Work

  • HER, Rauber et al., and Goyal et al. reuse trajectory information for goal-conditioned learning, whereas this work conditions on future state or return and does not act using the conditional policy.Prioritized sweeping can also be viewed as changing the sampling distribution using hindsight knowledge of TD errors.Source: §6 · p. 8
  • Temporal value transport [10] uses attention over memory to skip trajectory segments deemed irrelevant to obtained rewards, but the cited method is biased.Applying the paper's unbiased hindsight mechanism with attention-selected past states is proposed as future work.Source: §6 · pp. 8–9
  • Unlike variational intrinsic control [5], which uses inverse models to derive intrinsic rewards for policy training, this work uses an inverse model to determine an action’s influence on a future outcome.Other variance-reduction approaches use same-state rollouts, differentiable dynamics, or inferred latent causes, [9],.Source: §6 · p. 9

7 Closing

The closing section presents hindsight credit assignment as a new algorithm family and identifies deep reinforcement learning scalability as an open research direction.

7 Closing

  • The proposed algorithms explicitly address credit assignment as part of, or instead of, traditional value-function estimation.The paper reports that the estimators have new properties and empirically address some key credit-assignment issues; their scalability in deep reinforcement learning remains future work.Source: §7 · p. 9
  • The empirical validation supports addressing some key credit-assignment issues, while scalability in deep reinforcement learning remains an open research problem.The conclusion frames deep-RL scalability as future research rather than an established capability.Source: §7 · p. 9
  • The proposed family treats credit assignment as an explicit learning objective rather than relying only on traditional value-function estimation.The conclusion allows this treatment to be part of or instead of traditional estimation.Source: §7 · p. 9

A.1 Proof of Theorem 1

The proof excerpt invokes Bayes' rule to relate a future-state event to the initial action, then refers to the Q-function definition and equations in deriving Theorem 1.

A.1 Proof of Theorem 1

  • The lemma concerns a state y reachable under the policy and an initial action a that has nonzero policy probability.The excerpt provides this condition before introducing its probability relation.Source: §A.1 · p. 11
  • The proof invokes Bayes' rule to relate the probability of reaching state y after choosing action a to the posterior probability of that action given y.The excerpt then says the theorem proof starts from the Q-function definition and combines equations (9) and (10).Source: §A.1 · p. 11

A.2 Proof of Theorem 2

The proofs and variants rewrite value functions using hindsight-conditioned distributions. The algorithms train these distributions from trajectories and use them to update the policy.

A.2 Proof of Theorem 2

  • The Theorem 2 proof rewrites the value function using Bayes’ rule.The displayed passage gives two probability-factorized forms, but does not provide enough readable context to specify further implications.Source: §A.2 · p. 11

A.3 Proof of Theorem 3

  • The Theorem 3 proof uses an earlier result and identifies a third equality, but the supplied passages do not state the resulting formula.A return-version derivation is also introduced.Source: §A.3 · p. 12

B Other variants

  • The other variants obtain V- and Q-functions for state and return conditioning, with stated support conditions on the action and sampled state.The state-conditioned action distribution relates action likelihood given a future state to the likelihood of that state given the action.Source: B Other variants · pp. 12–13

C Time-Independent State-Conditional Case

  • With β = γ, the time-independent state-conditional formulation rewrites Q using a future-state-conditioned action distribution.Its timestep is sampled from a geometric distribution rather than fixed, unlike the original hβ definition.Source: C Time-Independent State-Conditional… · pp. 13–14

D Algorithms

  • The state-conditional algorithms train h from trajectory state pairs, then use hindsight-conditioned returns to update the policy across actions.The return-conditional version trains h from observed returns and uses its estimate in the policy gradient.Source: D Algorithms · p. 17

E Experiment Details

The delayed-effect discussion explains how state HCA can help when bootstrapping before the delayed outcome. The experiments also report hyperparameter choices and sensitivity.

E Experiment Details

  • Return HCA is sensitive to the number of bins, while all variants are robust to the learning-rate choice.For Fig. 4, the return version used nb = 3; elsewhere it used nb = 10.Source: E Experiment Details · p. 17
  • The baseline learning rate was selected from 0.1, 0.2, 0.3, and 0.4, while HCA learning rates were chosen informally.For Fig. 4, α = 0.3 and αb = 0.4.Source: E Experiment Details · p. 17

F Bootstrapping with state HCA

  • State HCA can discourage the bad action and encourage the good one when bootstrapping with n < T, because h updates faster than π.With interim V(y) < 0, the differences between h and π make the HCA returns negative for the bad action and positive for the good action; the effect persisted across tested learning rates and initializations.Source: F Bootstrapping with state HCA · p. 18

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