On this publish, I’ll introduce a reinforcement studying (RL) algorithm primarily based on an “various” paradigm: divide and conquer. In contrast to conventional strategies, this algorithm is not primarily based on temporal distinction (TD) studying (which has scalability challenges), and scales properly to long-horizon duties.
We will do Reinforcement Studying (RL) primarily based on divide and conquer, as an alternative of temporal distinction (TD) studying.
Downside setting: off-policy RL
Our drawback setting is off-policy RL. Let’s briefly overview what this implies.
There are two lessons 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 to throw away previous information every time we replace the coverage. Algorithms like PPO and GRPO (and coverage gradient strategies usually) belong to this class.
Off-policy RL means we don’t have this restriction: we are able to use any sort of information, together with previous expertise, human demonstrations, Web information, and so forth. So off-policy RL is extra basic 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, roboticsdialogue programs, healthcare, and many others.), we frequently haven’t any alternative however to make use of off-policy RL. That’s why it’s such an necessary drawback.
As of 2025, I feel we’ve fairly good recipes for scaling up on-policy RL (e.gPPO, GRPO, and their variants). Nevertheless, we nonetheless haven’t discovered a “scalable” off-policy RL algorithm that scales properly to complicated, 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 (ieQ-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 way of 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 publish should you’re concerned with extra particulars).
To mitigate this drawback, individuals have combined 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 fashion, we are able to scale 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 properly), 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 undergo from excessive variance and suboptimality. So we are able to’t simply set $n$ to a big worth, and have to rigorously tune it for every process.
Is there a basically completely different technique to remedy this drawback?
The “Third” Paradigm: Divide and Conquer
My declare is {that a} third paradigm in worth studying, divide and conquermight present a great 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 fashion, we are able to (in concept) 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 undergo 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 concept. The issue was that it wasn’t clear the best way to truly do that in apply… 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 have been capable of scale up divide-and-conquer worth studying to extremely complicated duties (so far as I do know, that is the primary such work!) no less than in a single necessary class of RL issues, goal-conditioned RL. Aim-conditioned RL goals to be taught a coverage that may attain any state from another state. This gives 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 surroundings’s transition graph, and $V$ is the worth operate related to the sparse reward $r(s, g) = 1(s = g)$. Intuitivelywhich 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 have been on the lookout for!
The issue
Nevertheless, there’s one drawback right here. The difficulty is that it’s unclear how to decide on the optimum subgoal $w$ in apply. 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 giant state areas, we are able to’t do that. Principally, because of this earlier works have struggled to scale up divide-and-conquer worth studying, although this concept has been round for many years (the truth is, 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 primary contribution of our work is a sensible resolution to this concern.
The answer
Right here’s our key concept: we prohibit the search house of $w$ to the states that seem within the dataset, particularly, those who lie between $s$ and $g$ within the dataset trajectory. Additionally, as an alternative of trying to find the optimum $textual content{argmax}_w$, we compute a “delicate” $textual content{argmax}$ utilizing expectile regression. Specifically, we decrease 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 all the state house. Second, we stop worth overestimation from the $max$ operator by as an alternative 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 complicated duties, we immediately 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 giant, 1B-sized datasets. These duties are extremely difficult: they require performing combinatorially complicated abilities throughout as much as 3,000 surroundings steps.
TRL achieves the perfect efficiency on extremely difficult, long-horizon duties.
The outcomes are fairly thrilling! In comparison with many robust baselines throughout completely different classes (TD, MC, quasimetric studying, and many others.), TRL achieves the perfect efficiency on most duties.
TRL matches the perfect, 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 completely different values of $n$, from $1$ (pure TD) to $infty$ (pure MC). The result’s very nice. TRL matches the perfect TD-$n$ on all duties, with no need to set $boldsymbol{n}$! That is precisely what we needed from the divide-and-conquer paradigm. By recursively splitting a trajectory into smaller ones, it could naturally deal with lengthy horizons, with out having to arbitrarily select the size of trajectory chunks.
The paper has loads of further experiments, analyses, and ablations. In the event you’re , take a look at 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 a very powerful query is the best way to prolong TRL to common, reward-based RL duties past goal-conditioned RL. Would common RL have an identical divide-and-conquer construction that we are able to exploit? I’m fairly optimistic about this, on condition that it’s doable to transform any reward-based RL process to a goal-conditioned one no less than in concept (see web page 40 of this e book).
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One other necessary problem is to take care of stochastic environments. The present model of TRL assumes deterministic dynamics, however many real-world environments are stochastic, primarily attributable to partial observability. For this, “stochastic” triangle inequalities would possibly present some hints.
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Virtually, I feel there’s nonetheless loads 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 scale back hyperparameters, additional stabilize coaching, and simplify the algorithm much more.
On the whole, I’m actually excited concerning the potential of the divide-and-conquer paradigm. I nonetheless suppose probably the most necessary 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 usually, is likely one of the strongest candidates towards this holy grail (by the way in which, I feel 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, corresponding to shortcut fashions, log-linear consideration, and recursive language fashions (and naturally, basic algorithms like quicksort, section 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 prefer to thank Kevin and Sergey for his or her useful suggestions on this publish.
This publish initially appeared on Seohong Park’s weblog.
