On this put up, I’ll introduce a reinforcement studying (RL) algorithm primarily based on an “various” paradigm: divide and conquer. Not like conventional strategies, this algorithm is not primarily based on temporal distinction (TD) studying (which has scalability challenges), and scales effectively to long-horizon duties.

We are able to do Reinforcement Studying (RL) primarily based on divide and conquer, as a substitute of temporal distinction (TD) studying.

Downside setting: off-policy RL

Our downside setting is off-policy RL. Let’s briefly assessment what this implies.

There are two courses of algorithms in RL: on-policy RL and off-policy RL. On-policy RL means we are able to solely use recent information collected by the present coverage. In different phrases, we’ve got to throw away outdated information every time we replace the coverage. Algorithms like PPO and GRPO (and coverage gradient strategies normally) belong to this class.

Off-policy RL means we don’t have this restriction: we are able to use any form of information, together with outdated expertise, human demonstrations, Web information, and so forth. So off-policy RL is extra normal and versatile than on-policy RL (and naturally more durable!). Q-learning is probably the most well-known off-policy RL algorithm. In domains the place information assortment is dear (e.g., robotics, dialogue programs, healthcare, and so on.), we frequently don’t have any alternative however to make use of off-policy RL. That’s why it’s such an vital downside.

As of 2025, I believe we’ve got moderately good recipes for scaling up on-policy RL (e.g., PPO, GRPO, and their variants). Nevertheless, we nonetheless haven’t discovered a “scalable” off-policy RL algorithm that scales effectively to advanced, long-horizon duties. Let me briefly clarify why.

Two paradigms in worth studying: Temporal Distinction (TD) and Monte Carlo (MC)

In off-policy RL, we sometimes prepare a worth perform utilizing temporal distinction (TD) studying (i.e., Q-learning), with the next Bellman replace rule:

[begin{aligned} Q(s, a) gets r + gamma max_{a’} Q(s’, a’), end{aligned}]

The issue is that this: the error within the subsequent worth $Q(s’, a’)$ propagates to the present worth $Q(s, a)$ by bootstrapping, and these errors accumulate over your complete horizon. That is principally what makes TD studying wrestle to scale to long-horizon duties (see this put up when you’re occupied with extra particulars).

To mitigate this downside, individuals have blended TD studying with Monte Carlo (MC) returns. For instance, we are able to do $n$-step TD studying (TD-$n$):

[begin{aligned} Q(s_t, a_t) gets sum_{i=0}^{n-1} gamma^i r_{t+i} + gamma^n max_{a’} Q(s_{t+n}, a’). end{aligned}]

Right here, we use the precise Monte Carlo return (from the dataset) for the primary $n$ steps, after which use the bootstrapped worth for the remainder of the horizon. This manner, we are able to cut back the variety of Bellman recursions by $n$ instances, so errors accumulate much less. Within the excessive case of $n = infty$, we recuperate pure Monte Carlo worth studying.

Whereas this can be a cheap resolution (and infrequently works effectively), it’s extremely unsatisfactory. First, it doesn’t basically remedy the error accumulation downside; it solely reduces the variety of Bellman recursions by a relentless issue ($n$). Second, as $n$ grows, we endure from excessive variance and suboptimality. So we are able to’t simply set $n$ to a big worth, and have to fastidiously tune it for every process.

Is there a basically totally different method to remedy this downside?

The “Third” Paradigm: Divide and Conquer

My declare is {that a} third paradigm in worth studying, divide and conquer, could present a super resolution to off-policy RL that scales to arbitrarily long-horizon duties.

Divide and conquer reduces the variety of Bellman recursions logarithmically.

The important thing concept of divide and conquer is to divide a trajectory into two equal-length segments, and mix their values to replace the worth of the complete trajectory. This manner, we are able to (in concept) cut back the variety of Bellman recursions logarithmically (not linearly!). Furthermore, it doesn’t require selecting a hyperparameter like $n$, and it doesn’t essentially endure from excessive variance or suboptimality, in contrast to $n$-step TD studying.

Conceptually, divide and conquer actually has all the great properties we wish in worth studying. So I’ve lengthy been enthusiastic about this high-level concept. The issue was that it wasn’t clear the right way to really do that in follow… till just lately.

A sensible algorithm

In a latest work co-led with Aditya, we made significant progress towards realizing and scaling up this concept. Particularly, we had been in a position to scale up divide-and-conquer worth studying to extremely advanced duties (so far as I do know, that is the primary such work!) at the least in a single vital class of RL issues, goal-conditioned RL. Aim-conditioned RL goals to be taught a coverage that may attain any state from every other state. This offers a pure divide-and-conquer construction. Let me clarify this.

The construction is as follows. Let’s first assume that the dynamics is deterministic, and denote the shortest path distance (“temporal distance”) between two states $s$ and $g$ as $d^*(s, g)$. Then, it satisfies the triangle inequality:

[begin{aligned} d^*(s, g) leq d^*(s, w) + d^*(w, g) end{aligned}]

for all $s, g, w in mathcal{S}$.

By way of values, we are able to equivalently translate this triangle inequality to the next “transitive” Bellman replace rule:

[begin{aligned}
V(s, g) gets begin{cases}
gamma^0 & text{if } s = g, \
gamma^1 & text{if } (s, g) in mathcal{E}, \
max_{w in mathcal{S}} V(s, w)V(w, g) & text{otherwise}
end{cases}
end{aligned}]

the place $mathcal{E}$ is the set of edges within the atmosphere’s transition graph, and $V$ is the worth perform related to the sparse reward $r(s, g) = 1(s = g)$. Intuitively, which means that we are able to replace the worth of $V(s, g)$ utilizing two “smaller” values: $V(s, w)$ and $V(w, g)$, supplied that $w$ is the optimum “midpoint” (subgoal) on the shortest path. That is precisely the divide-and-conquer worth replace rule that we had been in search of!

The issue

Nevertheless, there’s one downside right here. The difficulty is that it’s unclear how to decide on the optimum subgoal $w$ in follow. In tabular settings, we are able to merely enumerate all states to search out the optimum $w$ (that is basically the Floyd-Warshall shortest path algorithm). However in steady environments with massive state areas, we are able to’t do that. Principally, that is why earlier works have struggled to scale up divide-and-conquer worth studying, though this concept has been round for many years (in reality, it dates again to the very first work in goal-conditioned RL by Kaelbling (1993) – see our paper for an extra dialogue of associated works). The principle contribution of our work is a sensible resolution to this challenge.

The answer

Right here’s our key concept: we limit the search area of $w$ to the states that seem within the dataset, particularly, people who lie between $s$ and $g$ within the dataset trajectory. Additionally, as a substitute of looking for the optimum $textual content{argmax}_w$, we compute a “gentle” $textual content{argmax}$ utilizing expectile regression. Particularly, we reduce the next loss:

[begin{aligned} mathbb{E}left[ell^2_kappa (V(s_i, s_j) – bar{V}(s_i, s_k) bar{V}(s_k, s_j))right], finish{aligned}]

the place $bar{V}$ is the goal worth community, $ell^2_kappa$ is the expectile loss with an expectile $kappa$, and the expectation is taken over all $(s_i, s_k, s_j)$ tuples with $i leq ok leq j$ in a randomly sampled dataset trajectory.

This has two advantages. First, we don’t want to look over your complete state area. Second, we forestall worth overestimation from the $max$ operator by as a substitute utilizing the “softer” expectile regression. We name this algorithm Transitive RL (TRL). Try our paper for extra particulars and additional discussions!

Does it work effectively?

To see whether or not our technique scales effectively to advanced duties, we instantly evaluated TRL on among the most difficult duties in OGBench, a benchmark for offline goal-conditioned RL. We primarily used the toughest variations of humanoidmaze and puzzle duties with massive, 1B-sized datasets. These duties are extremely difficult: they require performing combinatorially advanced abilities throughout as much as 3,000 atmosphere steps.

TRL achieves one of the best efficiency on extremely difficult, long-horizon duties.

The outcomes are fairly thrilling! In comparison with many robust baselines throughout totally different classes (TD, MC, quasimetric studying, and so on.), TRL achieves one of the best efficiency on most duties.

TRL matches one of the best, individually tuned TD-$n$, while not having to set $boldsymbol{n}$.

That is my favourite plot. We in contrast TRL with $n$-step TD studying with totally different values of $n$, from $1$ (pure TD) to $infty$ (pure MC). The result’s very nice. TRL matches one of the best TD-$n$ on all duties, while not having to set $boldsymbol{n}$! That is precisely what we wished from the divide-and-conquer paradigm. By recursively splitting a trajectory into smaller ones, it could possibly naturally deal with lengthy horizons, with out having to arbitrarily select the size of trajectory chunks.

The paper has quite a lot of extra experiments, analyses, and ablations. In the event you’re , try our paper!

What’s subsequent?

On this put up, I shared some promising outcomes from our new divide-and-conquer worth studying algorithm, Transitive RL. That is just the start of the journey. There are various open questions and thrilling instructions to discover:

• Maybe an important query is the right way to prolong TRL to common, reward-based RL duties past goal-conditioned RL. Would common RL have the same divide-and-conquer construction that we are able to exploit? I’m fairly optimistic about this, provided that it’s potential to transform any reward-based RL process to a goal-conditioned one at the least in concept (see web page 40 of this guide).

• One other vital problem is to take care of stochastic environments. The present model of TRL assumes deterministic dynamics, however many real-world environments are stochastic, primarily as a consequence of partial observability. For this, “stochastic” triangle inequalities may present some hints.

• Virtually, I believe there may be nonetheless quite a lot of room to additional enhance TRL. For instance, we are able to discover higher methods to decide on subgoal candidates (past those from the identical trajectory), additional cut back hyperparameters, additional stabilize coaching, and simplify the algorithm much more.

On the whole, I’m actually excited in regards to the potential of the divide-and-conquer paradigm. I nonetheless suppose one of the vital issues in RL (and even in machine studying) is to discover a scalable off-policy RL algorithm. I don’t know what the ultimate resolution will appear like, however I do suppose divide and conquer, or recursive decision-making normally, is without doubt one of the strongest candidates towards this holy grail (by the way in which, I believe the opposite robust contenders are (1) model-based RL and (2) TD studying with some “magic” tips). Certainly, a number of latest works in different fields have proven the promise of recursion and divide-and-conquer methods, similar to shortcut fashions, log-linear consideration, and recursive language fashions (and naturally, basic algorithms like quicksort, phase timber, FFT, and so forth). I hope to see extra thrilling progress in scalable off-policy RL within the close to future!

Acknowledgments

I’d wish to thank Kevin and Sergey for his or her useful suggestions on this put up.

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This put up initially appeared on Seohong Park’s weblog.