On this publish, I’ll introduce a reinforcement studying (RL) algorithm based mostly on an “various” paradigm: divide and conquer. Not like conventional strategies, this algorithm is not based mostly on temporal distinction (TD) studying (which has scalability challenges), and scales properly to long-horizon duties.
We will do Reinforcement Studying (RL) based mostly on divide and conquer, as a substitute of temporal distinction (TD) studying.
Drawback setting: off-policy RL
Our drawback setting is off-policy RL. Let’s briefly evaluate what this implies.
There are two courses of algorithms in RL: on-policy RL and off-policy RL. On-policy RL means we will solely use contemporary information collected by the present coverage. In different phrases, we now have to throw away outdated information every time we replace the coverage. Algorithms like PPO and GRPO (and coverage gradient strategies generally) belong to this class.
Off-policy RL means we don’t have this restriction: we will use any form of information, together with outdated expertise, human demonstrations, Web information, and so forth.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 tougher!). Q-learning is probably the most well-known off-policy RL algorithm. In domains the place information assortment is pricey (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 essential drawback.
As of 2025, I believe we now have fairly good recipes for scaling up on-policy RL (e.g., PPO, GRPO, and their variants). Nonetheless, we nonetheless haven’t discovered a “scalable” off-policy RL algorithm that scales properly 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 usually prepare a worth operate 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 $Q(s, a)$ via bootstrapping, and these errors accumulate over all the horizon. That is principally what makes TD studying wrestle to scale to long-horizon duties (see this post if you happen to’re concerned about extra particulars).
To mitigate this drawback, folks have combined TD studying with Monte Carlo (MC) returns. For instance, we will 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 will scale back the variety of Bellman recursions by $n$ occasions, so errors accumulate much less. Within the excessive case of $n = infty$, we get better pure Monte Carlo worth studying.
Whereas it is a cheap resolution (and infrequently works well), it’s extremely unsatisfactory. First, it doesn’t basically remedy the error accumulation drawback; 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 will’t simply set $n$ to a big worth, and must rigorously tune it for every activity.
Is there a basically totally different approach to remedy this drawback?
The “Third” Paradigm: Divide and Conquer
My declare is {that a} third paradigm in worth studying, divide and conquer, might 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 thought 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 will (in principle) scale 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, not like $n$-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 thought. The issue was that it wasn’t clear the right way to truly do that in apply… till not too long ago.
A sensible algorithm
In a recent work co-led with Aditya, we made significant progress towards realizing and scaling up this concept. Particularly, we had been capable of scale up divide-and-conquer worth studying to extremely advanced duties (so far as I do know, that is the primary such work!) a minimum of in a single essential class of RL issues, goal-conditioned RL. Objective-conditioned RL goals to study 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 will 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 operate related to the sparse reward $r(s, g) = 1(s = g)$. Intuitively, because of this we will 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 on the lookout for!
The issue
Nonetheless, there’s one drawback right here. The problem is that it’s unclear how to decide on the optimum subgoal $w$ in apply. In tabular settings, we will merely enumerate all states to seek out the optimum $w$ (that is basically the Floyd-Warshall shortest path algorithm). However in steady environments with massive state areas, we will’t do that. Mainly, because of this earlier works have struggled to scale up divide-and-conquer worth studying, despite the fact that this concept has been round for many years (in truth, 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 difficulty.
The answer
Right here’s our key thought: we limit the search area of $w$ to the states that seem within the dataset, particularly, those 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 “delicate” $textual content{argmax}$ 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 $i leq okay leq j$ in a randomly sampled dataset trajectory.
This has two advantages. First, we don’t want to look over all the state area. Second, we stop worth overestimation from the $max$ operator by as a substitute utilizing the “softer” expectile regression. We name this algorithm Transitive RL (TRL). Take a look at our paper for extra particulars and additional discussions!
Does it work properly?
humanoidmaze
puzzle
To see whether or not our methodology scales properly to advanced duties, we immediately evaluated TRL on a few of 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-$n$, with no need 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, with no need 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 might probably naturally deal with lengthy horizons, with out having to arbitrarily select the size of trajectory chunks.
The paper has loads of extra experiments, analyses, and ablations. For those who’re , try our paper!
What’s subsequent?
On this publish, I shared some promising outcomes from our new divide-and-conquer worth studying algorithm, Transitive RL. That is only the start of the journey. There are various open questions and thrilling instructions to discover:
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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 will exploit? I’m fairly optimistic about this, on condition that it’s attainable to transform any reward-based RL activity to a goal-conditioned one a minimum of in principle (see web page 40 of this book).
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One other essential 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 result of partial observability. For this, “stochastic” triangle inequalities may present some hints.
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Virtually, I believe there’s nonetheless loads of room to additional enhance TRL. For instance, we will discover higher methods to decide on subgoal candidates (past those from the identical trajectory), additional scale back hyperparameters, additional stabilize coaching, and simplify the algorithm much more.
Basically, I’m actually excited concerning the potential of the divide-and-conquer paradigm. I still suppose one of the vital essential 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 to be, however I do suppose divide and conquer, or recursive decision-making generally, is among 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 current works in different fields have proven the promise of recursion and divide-and-conquer methods, akin to shortcut models, log-linear attention, and recursive language models (and naturally, basic algorithms like quicksort, phase bushes, FFT, and so forth). I hope to see extra thrilling progress in scalable off-policy RL within the close to future!
Acknowledgments
I’d prefer to thank Kevin and Sergey for his or her useful suggestions on this publish.
This publish initially appeared on Seohong Park’s blog.
