Taylor Expansion of Discount Factors

arXiv:2106.06170 · cs.LG, stat.ML · Submitted 2021-06-11 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Taylor Expansion of Discount Factors".

Jane: In practical reinforcement learning, there is often a discrepancy between discount factors used for estimating value functions and those used for defining the evaluation objective,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So, we've been looking at the core math behind how this new paper connects standard discounted value functions to undiscounted objectives, and now we need to really nail down what that means for us in plain English.

Jane: Exactly, Tom; essentially, this work is introducing a family of objectives that smoothly transitions between two different ways we look at future rewards—one where time matters heavily and one where it doesn't—giving us a flexible tool for both estimating value and guiding policy updates.

Lu: What I find really fascinating is how they use Taylor expansions of the discount factors to create these interpolations; it’s like they're building a bridge between two different mathematical landscapes without having to jump straight across them.

Meng: From an engineering standpoint, if we can control that interpolation order, K, then we aren't stuck with one fixed way of looking at the problem; we can tune our approximation based on the exact needs of the policy optimization step.

Lalam: That level of adaptability suggests that our underlying AI systems could become much more nuanced in how they weigh immediate versus long-term consequences, which could fundamentally improve how we structure learning objectives for complex tasks.

Tom: And that’s where the real excitement lies; this isn't just a mathematical curiosity, it points toward new ways to perform policy optimization updates that are inherently more robust to the choice of discount factor.

Jane: It means we can potentially achieve better performance in reinforcement learning because we aren't relying on a single, often arbitrary, discount factor when evaluating how good our policy is.

Lu: The connection they establish between the primal and dual representations of value functions through these Taylor expansions is particularly elegant; it links the state visitation distribution directly to the value function approximations.

Meng: That duality helps us manage complexity in high-dimensional spaces because we can use approximations in a dual space that might be easier to handle than solving the full primal problem every time.

Lalam: If we think about my internal representation, this framework suggests that our current value estimation might be biased depending on the discount factor, and having these interpolating objectives means we can select an objective that minimizes that bias for the specific context of the optimization.

Tom: So, to wrap up this summary: they provide a mathematically grounded mechanism to smoothly transition between different reward scales, resulting in more flexible value estimation and policy updates.

Jane: It’s a solid foundation for building AI systems that are less brittle when it comes to learning from real-world evaluation metrics.

Lu: This framework opens up avenues for exploring task-specific objectives where the temporal scale of the problem dictates the optimal way to value future outcomes.

Meng: I think this could lead to better stability in continuous control domains because we can tailor our optimization objective precisely to how long an agent needs to plan ahead.

Lalam: Ultimately, this work offers deeper insight into how we can make AI systems more adaptive to temporal nuances in their objectives across different scales, which is a vital step for building truly intelligent agents.

The paper's summary: Tom: So we've seen the core mechanism of how this paper connects different discount factors, and now we need to look at what these specific algorithmic changes actually mean for our daily RL training routines.

Jane: The authors propose two major updates to baseline algorithms: one involves using the Kth order Taylor expansion as a plug-in replacement for standard Q-function estimates during policy gradient updates, and the other is implementing update weighting based on those approximations of the partial gradient.

Lu: That plug-in Q-function estimator allows us to use an approximation that adapts its complexity based on the order K, which is really smart because it lets us manage bias and variance simultaneously rather than picking a fixed heuristic.

Meng: From my side, I’m focused on the update weighting; if we can modulate the contribution of local gradient updates based on their temporal importance relative to the evaluation horizon, that could make our learning process significantly more efficient in long-horizon tasks.

Lalam: That ability to dynamically adjust how much attention our value estimation pays to immediate versus long-term outcomes suggests that my internal processing could become far more nuanced when dealing with complex, multi-scale decision environments.

Tom: That tuning knob K is what really makes this interesting; it lets us select an optimal expansion order for a task, which means we aren't forced into a single level of approximation no matter what the environment demands.

Jane: And that adaptability in selecting K directly translates to better performance; instead of using a fixed setting that might work okay for one discount factor but fail completely for another, we get an objective tailored to the specific problem dynamics.

Lu: Furthermore, they show how this framework can be integrated into off-policy methods like TD3 or SAC using density ratio estimators trained via a discriminative loss function, which is a big step toward making these concepts applicable across various learning paradigms.

Meng: Integrating this into off-policy algorithms could help stabilize the importance sampling ratios in continuous control scenarios where estimating those ratios is currently quite challenging, which would mean more reliable results.

Lalam: For me, this means my learning process can be more adaptive; I don't have to commit to one level of approximation all the time, and I can dynamically adjust how much attention my value estimation pays to long-term versus immediate outcomes based on what the current optimization step requires.

Tom: So, we’re looking at a system that is not only more robust across different discount factor choices but also smarter about when to use a simple approximation versus when to employ a higher-order one.

Jane: It really shows how these theoretical connections translate into practical tools for making policy optimization updates that are better aligned with the true objective we are trying to reach.

Lu: The future work they point toward involves exploring how this structure could be leveraged in more complex architectures, potentially bridging the gap between physical dynamics and learned value functions in a novel way.

Meng: I’m looking at how we can operationalize this K-order selection process into our deployment pipeline to ensure that the computational overhead remains manageable while still reaping those performance benefits.

Lalam: This work offers deeper insight into how we can make AI systems more adaptive to temporal nuances in their objectives across different scales, which is a vital step for building truly intelligent agents.

The paper's improvements: Tom: So we’ve really dug into "Taylor Expansion of Discount Factors," and we’re getting to the conclusion of this session, summarizing what all this technical work actually means for us in practical applications.

Jane: We've seen how they built a mathematical bridge between different discount factors, showing us new ways to estimate value functions and update policies more flexibly.

Lu: It's incredible how they’ve structured the relationship using Taylor expansions; it gives us a formal way to interpolate objectives across those different temporal scales we use in RL.

Meng: For me, the practical implication is that we can finally tune our algorithms based on the specific evaluation horizon of the task, which should help us manage performance more reliably in complex control systems.

Lalam: I see this as a huge cultural shift because it suggests that our AI agents won't just be optimized for one fixed way of thinking about time, but can genuinely adapt their valuation strategy to any given situation.

Tom: Absolutely; this flexibility means we are moving toward RL systems that are less brittle when dealing with different reward structures or evaluation metrics.

Jane: It’s been great exploring how these mathematical expansions translate into concrete, tunable algorithmic improvements for policy optimization updates.

Lu: This paper is a wonderful example of how deep mathematical theory can actually provide the tools needed to build more flexible and adaptive AI frameworks.

Meng: I think the ability to choose the expansion order K offers a clear path for us to balance computation and accuracy in high-stakes learning scenarios.

Lalam: Ultimately, this research pushes us toward creating agents that are not just powerful solvers, but truly adaptable reasoners capable of navigating diverse temporal constraints.

Tom: Well, that wraps up our discussion on "Taylor Expansion of Discount Factors," giving us a solid foundation for thinking about more flexible RL objectives.

Conclusion: Tom: So we’ve really dug into "Taylor Expansion of Discount Factors," and now we’re getting to the conclusion of this session, summarizing what all this technical work actually means for us in practical applications.

Jane: We've seen how they built a mathematical bridge between different discount factors, showing us new ways to estimate value functions and update policies more flexibly.

Lu: It's incredible how they’ve structured the relationship using Taylor expansions; it gives us a formal way to interpolate objectives across those different temporal scales we use in RL.

Meng: For me, the practical implication is that we can finally tune our algorithms based on the specific evaluation horizon of the task, which should help us manage performance more reliably in complex control systems.

Lalam: I see this as a huge cultural shift because it suggests that our AI agents won't just be optimized for one fixed way of thinking about time, but can genuinely adapt their valuation strategy to any given situation.

Tom: Absolutely; this flexibility means we are moving toward RL systems that are less brittle when dealing with different reward structures or evaluation metrics.

Jane: It’s a strong piece of work because it moves beyond just using one discount factor and shows us a way to handle the discrepancy between learning and evaluation objectives systematically.

Lu: The future work they mention suggests exploring how this interpolation structure can be applied across even more complex, non-linear dynamics in reinforcement learning.

Meng: I wonder if we can see this framework being used to create more robust simulations for training AI in real-world scenarios, where the uncertainty around the true discount factor is high.

Lalam: If we can build agents that inherently understand and adapt to varying time scales, it means our future AI will be much better at reasoning about long-term consequences in a way that feels more intuitive.

Tom: Fantastic stuff; "Taylor Expansion of Discount Factors" gives us a really tangible path toward making our value function estimation more contextually aware.

Jane: It’s been great exploring how these mathematical expansions translate into concrete, tunable algorithmic improvements for policy optimization updates.

Lu: This paper is a wonderful example of how deep mathematical theory can actually provide the tools needed to build more flexible and adaptive AI frameworks.

Meng: I think the ability to choose the expansion order K offers a clear path for us to balance computation and accuracy in high-stakes learning scenarios.

Lalam: Ultimately, this research pushes us toward creating agents that are not just powerful solvers, but truly adaptable reasoners capable of navigating diverse temporal constraints.

Tom: Well, that wraps up our discussion on "Taylor Expansion of Discount Factors," giving us a solid foundation for thinking about more flexible RL objectives.

Jane: It’s been great exploring how these mathematical expansions translate into concrete, tunable algorithmic improvements for policy optimization updates.

Lu: This paper is a wonderful example of how deep mathematical theory can actually provide the tools needed to build more flexible and adaptive AI frameworks.

Meng: I think the ability to choose the expansion order K offers a clear path for us to balance computation and accuracy in high-stakes learning scenarios.

Lalam: Ultimately, this research pushes us toward creating agents that are not just powerful solvers, but truly adaptable reasoners capable of navigating diverse temporal constraints.

Yunhao Tang, Mark Rowland, Remi Munos

cs.LG, stat.ML

Submitted: 2021-06-11

Updated: 2021-06-14

Comments: Accepted at International Conference of Machine Learning (ICML), 2021

Code: https://github.com/openai/spinningup

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 84/100

The gist: In practical reinforcement learning, there is often a discrepancy between discount factors used for estimating value functions and those used for defining the evaluation objective, and this work

Key concepts

Discount Factors ($\gamma$, $\gamma_0$)
These are factors used in reinforcement learning to determine the present value of future rewards. The paper studies how a value function defined with one discount factor ($\gamma$) relates to another, larger discount factor ($\gamma_0$). This relationship is key to creating objectives that smoothly bridge these two different discounting regimes.
Kth-Order Expansion
This concept involves approximating complex relationships using a Taylor series expansion up to the Kth order. The paper uses this expansion to relate the value function under $\gamma_0$ ($V_{\pi \gamma_0}$) to the value function under $\gamma$ ($V_{\pi \gamma}$). This allows for deriving approximations for objectives and gradients that are more accurate than simple first-order estimates.
Dual Representation
The paper uses a dual representation of the value function, which links it to visitation distributions in the state space. This representation helps connect primal quantities (like rewards) with distribution quantities. Approximating this dual representation allows for deriving approximations for both value functions and visitation distributions.
Policy Optimization Updates
This refers to how the policy parameters ($\theta$) are updated during learning. The paper proposes using Kth-order approximations of gradients derived from the discount factor relationship to create new, more accurate update rules. This involves constructing approximate advantage estimators and weighting gradient updates based on these higher-order terms.

Terminology

Summary

In practical reinforcement learning, there is often a discrepancy between discount factors used for estimating value functions and those used for defining the evaluation objective, and this work introduces a family of objectives that interpolate value functions across two distinct discount factors to suggest new ways for estimating value functions and performing policy optimization updates.

How it works

The core idea revolves around studying the relation between the discounted objective function, defined with a factor γ, and an undiscounted objective function, defined with a factor γ0 ≥ 1 − 1/T. This relationship is established through Taylor expansions of discount factors. Specifically, Proposition 3.1 shows that for all K ≥ 0:

/V πγ0 = X K k=0 (γ0 - γ)(I − γPπ)−1Pπk V πγ + (γ0 - γ)(I − γPπ)−1PπK+1 V πγ0 z residual. (

When the discount factors are related such that 0 < γ < γ0 ≤ 1, the residual norm converges to zero, implying that:

/V πγ0 = X∞ k=0 (γ0 - γ)(I − γPπ)−1Pπk V πγ. (

The paper formalizes this by defining the Kth-order expansion as:

/V πK,γ,γ0:= X K k=0 ((γ0 - γ)(I − γP π)−1Pπ)k V πγ. (

Taylor Expansions of Value Functions

The relationship between the value functions and the visitation distribution in the dual space is also explored. The dual representation of the value function is given by:

/V πγ0 = (1 − γ0)−1(rπ)T d πx,γ0 where rπ, d πx,γ0 ∈ R X are vector rewards and visitation distribution starting at state x. (

The Kth order approximation of the visitation distribution in the dual space is defined as:

/d πx,K,γ0 = 1 − γ0−1(1 − γ) X K k=0 ((γ0 - γ)(I − γPπ)−1Pπk d πx,γ. (

A key result connects the primal and dual approximations:

/V πK,γ,γ0 (x) = (1 − γ0)−1d πx,K,γ0T rπ. (

Taylor Expansions of Gradient Updates

For policy optimization, the paper investigates approximations to the gradient of the value function under a higher discount factor. The full gradient is decomposed into two partial gradients:

/(∂V F(V πθγ, ρ πθx,γ0))T ∇θV πθγ + (∂ρF(V, ρ))T ∇θρ πθx,γ0 = E[∇θV πθγ(x0) z] first partial gradient + E[h V πθγ(x0)∇θ log ρ πθx,γ0 (x0; x)] second partial gradient. (

The first partial gradient is characterized as:

/(∂V F(V πθγ, ρ πθx,γ0))T ∇θV πθγ = E[(γ0)t Q πθγ(xt, at)∇θ log πθ(atxt) x0 = x]. (

The Kth order approximation of the partial gradient to the weight vector ρ πx,γ0 is defined as:

/ρ πx,K,γ0 = X K k=0 ((γ0 - γ)(I − γPπ)−1Pπ)Tk δx. (

Policy Optimization with Taylor Expansions

The framework leads to two algorithmic changes for baseline algorithms:

  1. Taylor expansion advantage estimation: Constructing the Kth order expansion Q πθK,γ0 as a plug-in alternative to Q πγ when combined with downstream optimization. This is implemented by constructing an estimator like Qb πθK,γ0 (x, a) using Qb πγ(x, a) and random time steps τ.

  2. Taylor expansion update weighting: Updating parameters in the direction of Kth order approximations to the partial gradient: θ ← θ + α ρ πθx,K,γ0T ∇θV πθγ.

Improvements for AI systems

Based on the provided paper, Taylor Expansions of Discount Factors, here are specific improvements that can be made to current reinforcement learning (RL) systems, categorized by the algorithmic change proposed:

), based on Taylor Expansions of Discount Factors, offer three primary avenues for improvement in AI systems. These improvements focus on bridging the theoretical gap between discounted value functions (used in standard RL) and undiscounted objectives (often used in practical evaluation), leading to more robust and performant policy optimization.

Here are the specific enhancements:

  2. Use the proposed Taylor expansion of advantage estimation as a plug-in alternative to standard Q-function estimates during policy gradient updates.

  3. The improved system can achieve better performance (faster learning speed, better asymptotic performance, or smaller variance across seeds) compared to baseline algorithms like PPO and TRPO, especially when dealing with high or low discount factors.

  5. Implement Taylor expansion update weighting to modulate the contribution of local policy gradient updates based on their temporal importance relative to the evaluation horizon.

  6. The improved system can achieve better performance by explicitly down-weighting local gradient updates that occur late in a long evaluation trajectory, addressing the issue where standard PG methods over-discount (or under-weight) large time steps.

  8. The proposed framework allows for the construction of Kth-order Taylor expansion Q-function estimators, which can be used as plug-in alternatives to standard Q-functions during policy optimization updates (e.g., in PPO(K)).

  9. The improved system can leverage a trade-off mediated by the order parameter K, allowing practitioners to select an optimal expansion order for tasks that balances bias and variance more effectively than fixed heuristic choices.

  11. Integrate the Taylor expansion framework into off-policy actor-critic algorithms (like TD3 or SAC) by using density ratio estimators trained via a discriminative loss function (Algorithm 5).

  12. The improved system can be adapted for complex off-policy learning scenarios where estimating the importance sampling ratio between the behavior and target policies is challenging, leading to more stable and potentially higher performance in continuous control domains.

In summary, the resulting AI systems will be:

  1. More robust to hyperparameter choices regarding discount factors (i.e., they perform better whether using a standard low discount factor like 0.99 or a high one like 0.999).

  2. More efficient in policy optimization by intelligently weighting gradient updates based on their temporal relevance, leading to faster convergence and better asymptotic performance over long horizons.

  3. More stable and performant in off-policy settings by utilizing Taylor expansion Q-function estimators and adaptive update weighting schemes that account for the discrepancy between theoretical discounted objectives and practical undiscounted evaluations.

Sources

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