Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret

arXiv:2104.11186 · cs.LG, stat.ML · Submitted 2021-04-22 · 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: "Stochastic Shortest Path".

Jane: Stochastic Shortest Path (SSP) learning involves an agent minimizing expected costs to reach a goal state in an environment where both transition dynamics and cost functions are initially unknown;

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

Paper summary: Tom: Welcome back to the show. We've been listening to some really fascinating papers lately, and today we're diving into something quite technical called "Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret." Jane, you’ve got the first round for us. What is this paper all about in a nutshell?

Jane: Well, Tom, it tackles the problem of learning in the stochastic shortest path setting, which is essentially when an agent tries to figure out the best sequence of actions to minimize accumulated costs before hitting a goal state. The main thesis here is that they designed a novel model-based algorithm called EB-SSP. They claim this algorithm manages to achieve a minimax regret rate that depends on Oe(B?√SAK), and they do this without needing any prior knowledge about things like the time-to-goal, T?, or the cost bounds, B?.

Lu: That sounds incredibly ambitious, Tom. Dealing with unknown dynamics and costs while aiming for that specific regret bound suggests a sophisticated way of handling uncertainty in online learning.

Meng: I'm curious about the practical side here. If an algorithm can achieve this minimax rate without knowing those prior parameters, how does that translate to real-world systems where we often lack perfect upfront information?

Lalam: From my perspective as a model, the core advance is showing how to build an optimistic SSP problem using EB-SSP and then having its value iteration scheme converge reliably. This convergence proof is a big piece of the puzzle for making these kinds of learning agents trustworthy.

Tom: Exactly, Lalam. So, the paper focuses on establishing three specific properties for any online SSP algorithm: minimax performance, parameter-free operation, and horizon-free regret bounds. That’s what sets this paper apart from previous work in this area.

Jane: Right, Tom? The paper lays out these desirable properties clearly: first, achieving the minimax regret rate Oe(B?√SAK), second, being parameter-free because it doesn't rely on knowing B? or T?, and third, having horizon-free regret bounds.

Paper summary: Lu: I find the focus on parameter-free learning really compelling. In complex environments, we often don't have perfect prior knowledge of the underlying cost structure or how long things will take to complete a task. Being able to learn robustly without that upfront data is huge for general AI agents.

Meng: Robustness is key, Lu. But the paper also shows how they handle the horizon dependency, which means the regret bound only depends logarithmically on T?, even when T? could be really large compared to B?. That addresses a major weakness in some other approaches.

Lalam: And looking at my own structure, I see that EB-SSP uses a specific mechanism involving perturbing empirical costs with an exploration bonus and biasing transitions toward the goal. This setup is what allows the associated value iteration scheme to converge reliably, as described in the abstract.

Tom: So, we're talking about a method that uses this specific perturbation strategy—the exploration bonus—to steer learning towards a convergence guarantee while simultaneously hitting that minimax benchmark without knowing B? or T?. It’s quite clever how they manage to decouple the performance from those unknown parameters.

Jane: That coupling is what makes it so interesting, Tom. The paper shows that this value-optimistic scheme works because of the way they carefully skew the empirical transitions and costs over time. This is a model-based approach that learns its own necessary structure incrementally, which is smart for online learning scenarios.

Lu: I think the novelty lies in how they manage to induce an optimistic SSP problem whose associated value iteration scheme is guaranteed to converge using these specific perturbations. That convergence guarantee under those constraints is what makes this approach feasible in practice for many problems.

Meng: Practically speaking, when we deploy these agents, we need efficiency. The computational complexity analysis mentioned suggests it's polynomially bounded and near-linear in T, even with the estimation of B?. That’s good news for scaling up these learning systems on larger state spaces.

Paper summary: Lalam: Efficiency is important because it dictates how quickly we can deploy these improvements into our existing AI culture. If the complexity remains manageable, we can start seeing these performance gains in real applications faster than if the overhead was too high.

Tom: Speaking of performance, let's move on to what this all means for us in conclusion. We're talking about "Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret." This paper addresses the fundamental challenge of learning under uncertainty in SSP settings by providing an algorithm that hits a theoretical lower bound while being completely parameter-free.

Jane: Simply put, the implication is that we can build agents that learn to navigate complex, unknown cost landscapes very effectively without needing to pre-calculate those hard metrics like the total expected cost B? or the expected time T?. It gives us a powerful tool for situations where upfront knowledge is simply not available.

Lu: For me, it opens up possibilities for designing AI agents that are much more adaptable to dynamic environments where costs and goals might shift unexpectedly over time. This flexibility is what I'm most excited about; the potential for truly general learning systems increases significantly with this kind of algorithm.

Meng: From an engineering standpoint, this means we can deploy more powerful agents in areas like adaptive control or resource allocation where the optimal path isn't known beforehand but needs to be found through interaction alone. It moves us closer to systems that react intelligently rather than just following pre-programmed rules.

Lalam: I think the cultural impact lies in developing AI systems that exhibit this kind of self-directed, robust learning capability. If we can build agents that handle uncertainty without needing exhaustive prior knowledge, it fundamentally changes how we think about autonomous decision-making within our AI culture.

Tom: It sounds like a paper that really pushes the boundaries of what's possible in online learning for sequential decision-making problems. So, to wrap up this discussion on "Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret," this work suggests a path toward much more general and robust AI agents capable of handling high uncertainty without needing extensive prior knowledge.

Conclusion: Tom: So we've been deep in the weeds of EB-SSP, but now we need to wrap up what this whole paper is actually about and what it means for us out there.

Jane: Right, Tom, so essentially, the paper is introducing a method called EB-SSP that lets an AI agent learn to find the best path through a maze without needing any prior knowledge about the cost structure or how long those paths take.

Lu: It's really about achieving that minimax performance level while staying parameter-free and tackling that tricky horizon-free regret issue, which is a big deal theoretically.

Meng: From an engineering standpoint, it means we could deploy agents in environments where we don't have perfect upfront data to find the optimal sequence of moves.

Lalam: I see this as a major step forward in building more adaptable AI systems that can learn robustly on the fly without needing pre-programmed constraints.

Tom: Exactly, and thinking about the title, "Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret," it really sums up these three big wins of this algorithm.

Jane: That title captures the core idea perfectly; it's about finding that optimal path in a random setting without knowing the cost bounds or the time constraints upfront.

Lu: The minimax regret bound is what ties everything together, showing we can reach a performance level close to the best possible performance achievable by any policy.

Meng: And for me, the parameter-free aspect is critical because it lets us apply this to so many diverse problems where defining those prior knowledge parameters B? or T? would be practically impossible.

Lalam: It really speaks to how we can build AI that is more general and less brittle when faced with unpredictable real-world costs and dynamics.

Tom: So, what does this all mean for the future of AI deployment and decision-making in complex, uncertain settings?

Jane: It means we are moving toward agents that can handle learning in highly stochastic environments much more flexibly than before.

Lu: I think this approach opens up new avenues for designing complex AI systems that operate with inherent uncertainty rather than requiring perfect information.

Meng: For practical applications, it suggests we can build more resilient automated systems that don't fail when the environment deviates from our initial assumptions about costs or time.

Lalam: It pushes us to think about how AI can develop its own useful models of the world while still learning effectively in real-time.

Jean Tarbouriech, Runlong Zhou, Simon S. Du, Matteo Pirotta, Michal Valko, Alessandro Lazaric

Facebook AI Research & Inria Lille · Tsinghua University · University of Washington & Facebook AI Research · Facebook AI Research Paris (DeepMind) · Facebook AI Research Paris

cs.LG, stat.ML

Submitted: 2021-04-22

Updated: 2021-12-10

Comments: NeurIPS 2021

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

Importance score: 89/100

The gist: Stochastic Shortest Path (SSP) learning involves an agent minimizing expected costs to reach a goal state in an environment where both transition dynamics and cost functions are initially unknown;

Key concepts

Stochastic Shortest Path (SSP): This is a problem where an agent tries to find the best sequence of actions to minimize expected costs while moving from a starting state to a specific goal state in an uncertain environment. The agent learns by interacting with this environment.
Minimax Regret: This measures how much worse the algorithm's performance is compared to the absolute best possible policy, regardless of what the true unknown parameters (like total expected cost) are. Achieving a minimax bound means the performance is robust against these unknowns.
Horizon-Free Regret: This property means that even if you don't know exactly how long it will take to reach the goal (T?), the algorithm's regret still stays small, depending only on a logarithm of T?. This makes the learning process more practical because it doesn't require knowing a fixed time horizon beforehand.

Terminology

Regret — 137× in this paper · explained in 3 episodes
Stochastic Shortest Path — 22× in this paper
Minimax Regret — 12× in this paper · explained in 2 episodes
Reinforcement Learning — 9× in this paper · explained in 4 episodes
Core — 5× in this paper · explained in 2 episodes
Q-learning — 4× in this paper · explained in 3 episodes
Sample Complexity — 4× in this paper · explained in 15 episodes
AI Agents — 3× in this paper · explained in 2 episodes
Action Space — 2× in this paper · explained in 1 episode
Generative Model — 2× in this paper · explained in 2 episodes
Accuracy — 1× in this paper · explained in 1 episode
Benchmark — 1× in this paper · explained in 2 episodes
Gap-Dependent Regret — 1× in this paper · explained in 2 episodes
Generalization — 1× in this paper · explained in 13 episodes
Regularization — 1× in this paper · explained in 3 episodes
Reinforcement Learning (RL) — 1× in this paper · explained in 24 episodes
Robustness — 1× in this paper · explained in 5 episodes

Summary

Stochastic Shortest Path (SSP) learning involves an agent minimizing expected costs to reach a goal state in an environment where both transition dynamics and cost functions are initially unknown; this paper introduces EB-SSP, a novel model-based algorithm that achieves the minimax regret rate of Oe(B?√SAK) while being parameter-free and achieving horizon-free regret bounds.

Core Problem and Desired Properties

The study focuses on online learning in the SSP setting, where the agent seeks to achieve a performance as close as possible to the optimal policy π, measured by low regret. The paper identifies three desirable properties for such an algorithm:

Minimax: The regret is bounded by Oe(B?√SAK), where B? bounds the total expected cost of the optimal policy starting from any state.
Parameter-free: The algorithm relies neither on T? (expected time-to-goal) nor B? prior knowledge.
Horizon-free: Regret depends only logarithmically on T?, even when T? may be arbitrarily large relative to B?.

Algorithm EB-SSP and Optimism

The proposed algorithm, EB-SSP (Exploration Bonus for SSP), is a value-optimistic scheme designed to compute optimistic policies efficiently. It achieves this by:

(i) Perturbing the empirical costs with an exploration bonus:

(ii) Biasing the empirical transitions towards reaching the goal from each state-action pair with positive probability.

The algorithm decays this bias over time in a way that it only contributes to a lower-order regret term. The core mechanism involves a doubling update framework, similar to MVP, but performing policy updates instantaneously when the doubling condition is met.

Regret Analysis and Key Results

The analysis establishes several significant results regarding the performance of EB-SSP:

(i) Minimax Regret:

The algorithm achieves the minimax regret rate of Oe(B?√SAK) while being parameter-free, bypassing the need to know T? or B? prior knowledge.

(ii) Horizon-Free Bounds:

EB-SSP is shown to be the first algorithm to achieve horizon-free regret in various cases: i) positive costs, ii) no almost-sure zero-cost cycles, and iii) the general cost case when an order-accurate estimate of T? is available. For general costs with an order-accurate estimate of T?, the regret is bounded as Oe(B?√SAK + B?S2A).

(iii) Regret Decomposition:

The total regret RK is decomposed into three parts: X1 (error on optimistic V-values), X2 (Bellman error), and X3 (cost estimation error). The final bound for the general case with an order-accurate estimate of T? is shown to be:

RK = O(B?√SAK log B?T SAδ + B?S2A log2)

Parameter-Free Extensions

To address the challenge of unknown B?, a parameter-free version, EB-SSP (Alg. 2), is introduced. This version uses a proxy estimate Be, which is initialized to 1 and increased based on episode progress or cumulative cost thresholds. The algorithm employs two main increment strategies for Be:

(i) Episode-driven increment of Be:

Be ← max[Be, √k/(S3A1/2)] at the beginning of each new episode k. This ensures that Be will eventually hold that Be ≥ B? for large enough episodes.

(ii) Doubling increment of Be:

Be ← 2Be whenever a phase terminates due to exceeding a cumulative cost threshold or exceeding the VISGO range (kV(i)k∞ > Be). This allows the agent to become aware that its estimate is too small and thus doubles it.

Computational Efficiency

The computational complexity of EB-SSP is analyzed based on the total number of VISGO procedure executions, which are bounded by O(SA log T). The total computational complexity for EB-SSP is derived as O(T S2A · SA log(B?T) · SA log T), which is polynomially bounded and near-linear in T. This efficiency is maintained even when accounting for the complexity introduced by the unknown B? estimation process.

Alternative Assumptions

The paper explores alternative assumptions to guarantee horizon-free bounds:

(i) Positive Costs (Assumption 4):

Under the assumption that all costs are lower bounded by a constant cmin > 0, EB-SSP achieves a nearly minimax and horizon-free bound, where T? is bounded by T? ≤ B?/cmin.

Improvements for AI systems

As a fastidious researcher, I have analyzed this paper, Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret, by Tarbouriech et al. The core contribution is the algorithm EB-SSP (Exploration Bonus for SSP), which achieves minimax regret bounds while being parameter-free and nearly horizon-free in the Stochastic Shortest Path (SSP) setting.

Based on this research, here are specific improvements to AI systems and what those systems can achieve:

) 1. Robust, Goal-Oriented Navigation in Unknown Environments

By applying EB-SSP to navigation tasks (e.g., Mujoco mazes or complex robotic path planning), the resulting system will exhibit:

Specific capability: The ability to find the optimal path from any starting state to a goal, even when the exact cost function or transition dynamics are initially unknown.
Improvement over standard RL: Unlike algorithms that might get stuck in local optima due to poor initial estimates of costs (as seen in standard value iteration), EB-SSP's exploration bonus ensures that policies are proper (guaranteed to reach the goal with probability 1), leading to significantly lower cumulative cost regret.

) 2. Parameter-Free Policy Learning for Complex MDPs

The algorithm is parameter-free, meaning it does not require prior knowledge of the optimal time-to-goal estimate or the total expected cost bound. This allows for:

Specific capability: Deployment in real-world scenarios where environment models (transition probabilities and costs) are highly complex, non-stationary, or only partially observable. The agent learns the necessary exploration strategy (via the exploration bonus) without needing a pre-computed horizon or cost scale.

) 3. Horizon-Free Performance in Long-Term Planning

The paper demonstrates a nearly horizon-free regret bound with respect to the optimal time-to-goal estimate, even when instantaneous costs are very small (i.e., the gap between optimal cost and time is large). This leads to:

Specific capability: Superior long-term planning in environments where reaching the goal might take an arbitrarily long, unpredictable time. The system's performance will not degrade significantly just because the expected time-to-goal is very large or poorly estimated, a common failure point for finite-horizon planners.

) 4. Adaptive and Sample-Efficient Exploration Strategies

The VISGO procedure, which computes optimistic Q-values by perturbing transitions towards the goal (using a decay schedule), provides a highly informed exploration mechanism:

Specific capability: The system can dynamically adjust its exploration strategy based on empirical data density. When it visits a state-action pair infrequently, the bonus pushes the agent toward that pair to gather more information; as visitation increases, this bonus decays, allowing for exploitation of the learned optimal policy. This makes learning extremely sample-efficient compared to methods that explore uniformly or rely only on simple count statistics.

) 5. Guaranteed Computational Efficiency

The analysis shows a near-linear time complexity relative to the total accumulated time (T) within K episodes, i.e., O(T S2A log(B?T)). This ensures:

Specific capability: The system can operate in real-time or on large state spaces (S) for extended periods without computational bottlenecks, making it suitable for high-frequency decision-making in complex operational settings where the total time spent learning is significant.

In summary, the improved AI system will be a navigation agent that learns optimal goal-reaching policies in unknown environments with minimal regret and maximal sample efficiency, capable of handling long trajectories and complex cost structures without needing any prior knowledge of those bounds.