Benchmarking Tabular Reinforcement Studying Algorithms

posts, we explored Half I of the seminal guide Reinforcement Studying by Sutton and Barto [1] (*). In that part, we delved into the three elementary methods underlying practically each trendy Reinforcement Studying (RL) algorithm: Dynamic Programming (DP), Monte Carlo strategies (MC), and Temporal Distinction Studying (TD). We not solely mentioned algorithms from every discipline in depth but in addition carried out each in Python.

Half I of the guide focuses on tabular resolution strategies — approaches suited to issues sufficiently small to be represented in desk kind. As an example, with Q-learning, we are able to compute and retailer an entire Q-table containing each potential state-action pair. In distinction, Half II of Sutton’s guide tackles approximate resolution strategies. When state and motion areas change into too giant — and even infinite — we should generalize. Contemplate the problem of taking part in Atari video games: the state area is just too huge to mannequin exhaustively. As an alternative, deep neural networks are used to compress the state right into a latent vector, which then serves as the idea for an approximated worth operate [2].

Whereas we’ll enterprise into Half II in upcoming posts, I’m excited to announce a brand new sequence: we’ll benchmark all of the algorithms launched in Half I in opposition to each other. This put up serves each as a abstract and an introduction to our benchmarking framework. We’ll consider every algorithm based mostly on how rapidly and successfully it could actually clear up more and more bigger Gridworld environments. In future posts, we plan to increase our experiments to tougher eventualities, equivalent to two-player video games, the place the variations between these strategies shall be much more obvious.

Summarized, on this put up, we’ll:

Introduce the benchmark job and focus on our comparability standards.
Present a chapter-by-chapter abstract of the strategies launched in Sutton’s guide together with preliminary benchmarking outcomes.
Establish the best-performing strategies from every group and deploy them in a larger-scale benchmarking experiment.

Desk of Contents

Introducing the Benchmark Job and Experiment Planning

On this put up we’ll work with the Gymnasium [3] atmosphere “Gridworld”. It’s basically a maze-finding job, wherein the agent has to get from the top-left nook of the maze to the bottom-right tile (the current) — with out falling into any of the icy lakes:

The state area is a quantity between 0 and N — 1, the place N is the maximal variety of tiles (16 in our case). There are 4 actions the agent can execute in each step: going left, proper, up or down. Reaching the purpose yields reward 1, falling into the lake ends the episode with none reward.

The great factor about this atmosphere is that one can generate random worlds of arbitrary measurement. Thus, what we’ll do with all strategies, is plot the variety of steps / updates wanted to resolve the atmosphere versus the atmosphere measurement. In actual fact, Sutton does this in some elements of the guide, too, so we are able to confer with that.

Preliminaries

I’d like to begin with some basic notes — hoping these will information you thru my thought course of.

It isn’t straightforward to check algorithms in a “truthful” means. When implementing the algorithms I primarily regarded for correctness, but in addition simplicity — that’s, I wished readers to simply be capable to join the Python code with pseudocode from Sutton’s guide. For “actual” use circumstances, one would absolutely optimize the code extra, and in addition use a big array of methods frequent in RL, equivalent to utilizing decaying exploration, optimistic initialization, studying fee tuning, and extra. Additional, one would take nice care to tune the hyperparameters of the deployed algorithm.

Utilized RL Tips

As a result of giant variety of algorithms underneath investigation, I can not do that right here. As an alternative, I recognized two vital mechanisms, and measured their effectiveness on an algorithm recognized to work fairly nicely: Q-learning. These are:

Intermediate rewards: as an alternative of solely rewarding the agent for reaching the purpose, we reward it for progress alongside the maze. We quantify this by the (normalized) distinction in x and y coordinates between present and former state. This makes use of the truth that the purpose in every Gridworld atmosphere is all the time on the backside proper, and thus greater x / y coordinates normally are higher (though one may nonetheless get “caught”, in case an icy lake is between the agent and the purpose). Since this distinction is normalized by the variety of states, its contribution is small, s.t. it doesn’t overshadow the reward of reaching the purpose.
Decaying exploration: all through this put up sequence, the exploration-exploration dilemma got here up regularly. It describes the trade-off of exploiting states / actions already recognized to be good, and exploring much less explored states — doubtlessly discovering even higher options, on the threat of losing time in much less optimum areas. One frequent method addressing that is to decay exploration: beginning with a excessive quantity of exploration early, and slowly decaying this all the way down to decrease ranges. We do that by linearly decaying ε from 1 (100% random exploration) to 0.05 over 1000 steps.

Let’s take a look at how Q-learning performs with these methods:

As we are able to see, within the baseline setup the variety of steps wanted rapidly grows, and reaches the maximal variety of allowed steps (100.000 ) — which means the algorithm didn’t clear up the atmosphere within the allotted variety of steps. Additionally decaying ε alone didn’t contribute a lot. Nonetheless, including intermediate rewards proved to be extraordinarily efficient — and the mix of this and decaying ε carried out greatest.

Thus, for many strategies to come back we begin with the “naïve” atmosphere, the baseline implementation. Later we present outcomes for the “improved” atmosphere consisting of intermediate rewards and decaying exploration.

Comparability Standards

As was seen within the earlier part I selected the variety of steps wanted till the discovered coverage solves the Gridworld atmosphere because the default means of evaluating strategies. This appears a bit extra truthful than simply measuring elapsed time, since time is dependent upon the concrete implementation (much like the idea of Huge O notation)— and, as talked about above, I didn’t optimize for pace. Nonetheless, it is very important be aware that additionally steps might be deceptive, as e.g. one step in DP strategies comprises a loop over all states, whereas one step in MC and TD strategies is the era in a single episode (truly for TD strategies we normally rely one step as one worth replace, i.e. an episode era consists of a number of steps — nevertheless I made this extra akin to MC strategies on goal). Resulting from this we additionally present elapsed time typically.

Experiment Construction

To scale back the variance, for every Gridworld measurement we run every technique 3 times, after which save the bottom variety of steps wanted.

The code required to run all following benchmarking might be discovered on GitHub.

Recap and Benchmarking of All Algorithms

With the basics out of the best way, let’s correctly get began. On this part we’ll recap all launched algorithms from Half I of Sutton’s guide. Additional, we’ll benchmark them in opposition to one another on the beforehand launched Gridworld job.

Dynamic Programming

We begin with Chapter 4 of Sutton’s guide, describing strategies from DP. These might be utilized to all kinds of issues, all the time constructing on the precept of iteratively constructing bigger options from smaller subproblems. Within the context of RL, DP strategies keep a Q-table which is stuffed out incrementally. For this, we require a mannequin of the atmosphere, after which, utilizing this mannequin, replace the anticipated worth of states or state-action pairs relying on the potential successor states. Sutton introduces two strategies we picked up in our corresponding put up: coverage and worth iteration.

Let’s begin with coverage iteration. This consists of two interleaved steps, particularly coverage analysis and coverage enchancment. Coverage analysis makes use of DP to — because the identify says — consider the present coverage: we incrementally replace the state estimates through the use of the mannequin and coverage. Subsequent comes coverage enchancment, which employs a elementary idea of RL: based on the coverage enchancment theorem, any coverage will get higher when altering the anticipated motion in a single state to a greater motion. Following this, we assemble the brand new coverage from the Q-table in grasping style. That is repeated, till the coverage has converged.

The corresponding pseudocode seems to be as follows:

Let’s come to worth iteration. That is similar to coverage iteration, however with a easy, but essential modification: in each loop, just one step of coverage analysis is run. It may be proven that this nonetheless converges to the optimum coverage, and total does so quicker than coverage iteration:

For extra particulars, right here’s my corresponding put up about DP strategies.

Outcomes

Now it’s time to see what these first two algorithms are actually fabricated from. We run the experiment sketched above, and get the next plot:

Each strategies are in a position to clear up all created Gridworld sizes within the minimal variety of steps, 100. Shocking? Nicely, this truly reveals each a energy and in addition to a weak spot of DP strategies, as concurrently of our methodology: DP strategies are “thorough”, they require an entire mannequin of the world, after which iterate over all states — yielding an excellent resolution with only a few passes over all states. Nonetheless, which means all states must be estimated until convergence — although a few of them is likely to be a lot much less fascinating — and this scales fairly badly with the atmosphere measurement. In actual fact, one measured step right here comprises a full run over all states — indicating that for these strategies time is a greater measure.

For this, we get the next graph:

Now, we are able to see enhance in compute wanted w.r.t. to the variety of states. And, we are able to additionally see that, as claimed, worth iteration converges a lot quicker, and scales a lot better. Observe that the x-axis labels denote n, with the Gridworld having a measurement of n x n.

Monte Carlo Strategies

Subsequent in our put up sequence on RL we coated MC strategies. These can study from expertise alone, i.e. one can run them in any form of atmosphere, with out having a mannequin of it — which is a shocking realization, and really helpful: typically, we don’t have this mannequin, different occasions, it will be too advanced and impractical to make use of. Contemplate the sport of Blackjack: whereas we are able to actually mannequin all potential outcomes and corresponding chances, it’s a very tedious job — and studying to play by simply doing that may be a very tempting thought. Resulting from not utilizing a mannequin, MC strategies are unbiased — however on the draw back their expectation has a excessive variance.

One concern when implementing these strategies is ensuring that each one state-action pairs are constantly visited, and thus up to date. Resulting from not having a mannequin, we can not merely iterate over all potential combos (evaluate e.g. DP strategies), however (in a means) randomly discover the atmosphere. If on account of this we missed some states fully, we’d free theoretical convergence ensures, which might translate into observe.

A method of satisfying that is the exploring begins assumption (ES): we begin every episode in a random state and select the primary motion randomly, too. Aside from that, MC strategies might be carried out relatively merely: we merely play out full episodes, and set the anticipated worth of state-action pairs to the common obtained reward.

MC with ES seems to be as follows:

To take away the belief of ES, we are able to resort to 2 lessons of algorithms: on- and off-policy strategies. Let’s begin with the on-policy one.

That is truly not too totally different from the ES algorithm, we merely use an ε-greedy coverage for producing episodes. That’s, we take away the belief of ES and use a “comfortable” as an alternative of a “arduous” coverage for producing episodes: the used coverage in each iteration is just not totally grasping, however ε-greedy — which ensures that within the restrict we see all potential state-action pairs:

Off-policy strategies comply with the concept of splitting exploration and studying in two insurance policies. We keep a coverage π, which we wish to optimize, and a habits coverage, b.

Nonetheless, we are able to’t merely use b in every single place in our algorithm. When producing an episode and computing returns, we receive:

I.e., the ensuing worth is the anticipated worth of b, not π.

That is the place significance sampling is available in. We are able to repair this expectation with the best ratio:

This ratio is outlined by:

In our case, we receive the next method:

(Observe that this makes use of weighted significance sampling, as an alternative of “naïve” significance sampling.)

We may after all compute these ratios naively in each step. Nonetheless, Sutton introduces a intelligent scheme updating these values (denoted by W) incrementally, which is far more environment friendly. In actual fact, in my unique put up I confirmed the naive model, too — I consider this helps with understanding. Nonetheless, since right here we primarily care about benchmarking, and the “naïve” and the “incremental” model are similar, as much as efficiency — we right here solely checklist the marginally extra advanced incremental model.

In pseudocode the corresponding algorithm seems to be as follows:

Observe that, against our preliminary put up introducing these strategies, the place the habits coverage was merely randomly, right here we decide a greater one — particularly an ε-greedy one w.r.t. to the present Q-table.

For extra particulars right here’s my corresponding put up on MC strategies.

Outcomes

With that, let’s evaluate these three algorithms on small Gridworld environments. Observe that one step right here denotes one full episode generated:

We observe off-policy MC to already trip at a Gridword measurement of 5×5, and, although MC with ES and on-policy MC carry out higher, additionally they begin to battle with bigger sizes.

This is likely to be considerably stunning, and disappointing for MC followers. Don’t fear, we’ll handle to spice up this — nevertheless it reveals a weak spot of those algorithms: in “giant” environments with sparse rewards, MC strategies principally need to hope to come upon the purpose by likelihood — which decreases exponentially with the scale of the atmosphere.

Thus, let’s try to make the duty simpler for the mannequin, and use the beforehand launched methods empirically discovered to assist efficiency of TD-learning: including intermediate rewards, and ε-decay — our “improved” setup.

In actual fact, with this all strategies fare a lot better:

Nonetheless, now MC ES is inflicting issues. Thus, let’s put this apart and proceed with out it: ES anyhow was a theoretical idea on the best way of growing MC strategies, and clunky to make use of / implement (some would possibly bear in mind how I carried out having the atmosphere begin in random states …):

Right here, no less than we get near the outcomes of DP. Observe that I capped the maximal variety of steps to 100.000, so every time this quantity reveals up within the graph it implies that the algorithm couldn’t clear up this atmosphere within the given step restrict. On-policy MC truly appears to carry out rather well, the variety of steps wanted barely will increase— however off-policy MC appears to carry out worse.

Dialogue

To me, MC strategies carry out surprisingly nicely — since they basically stumble across the atmosphere randomly at first, hoping to seek out the purpose by exploration alone. Nonetheless, after all this isn’t totally true — their efficiency (talking of on-policy MC) turns into actually good solely after enabling intermediate rewards — which information the mannequin in the direction of the purpose. On this setup it appears MC strategies carry out rather well — one potential purpose being that they’re unbiased — and fewer delicate to hyperparameter tuning and co.

Temporal-Distinction Studying

Let’s come to TD strategies. These might be seen as combining the strengths of each approaches beforehand launched: much like MC, they don’t want a mannequin of the atmosphere — however nonetheless they construct upon earlier estimates, they bootstrap — as in DP.

Let’s recap DP and MC fashions:

DP strategies flip the Bellman equation into an replace rule, and compute the worth of a state based mostly on the estimated values of its successor states:

MC strategies, however, play out full episodes after which replace their worth estimates based mostly on the noticed return:

TD strategies mix these two concepts. They play out full episodes, however after each step replace worth estimates with the noticed return, and the earlier estimate:

A number of the most elementary RL algorithms stem from this discipline — and we’ll focus on them within the following.

Let’s start with Sarsa. First, we modify above launched replace rule to work with state-action pairs:

With this, Sarsa is definitely launched relatively rapidly: we play episodes, and replace values following our present coverage. The identify comes from the tuples used within the updates:

In pseudocode this seems to be as follows:

Subsequent up now we have Q-learning. That is similar to Sarsa, with one key distinction: it’s an off-policy algorithm. As an alternative of merely following the executed transition throughout the replace, we take the utmost Q-value of all successor states:

You possibly can image this as making a habits coverage b, which is the same as π, besides being grasping within the transitions underneath query.

The pseudocode seems to be like this:

One other algorithm is Anticipated Sarsa, which (you guessed it) — is an extension of Sarsa. As an alternative of following the one transition executed by the coverage, we account for all potential successor states, and weigh them by how possible they’re given the present coverage:

The final algorithm on this chapter is an extension of Q-learning. Q-learning suffers from an issue often known as maximization bias: because it makes use of a most over anticipated values, the ensuing estimate can have optimistic bias. We are able to deal with this through the use of two Q-tables: for every replace we use one for choosing a value-maximizing motion, and the opposite for computing the replace goal. Which is used the place is decided by a coin flip. The algorithm is named Double Q-learning:

Outcomes

Let’s take a look on the outcomes, beginning with the naïve atmosphere:

We are able to see that each Q-learning strategies begin to get issues with Gridworld sizes of 11 x 11.

Thus let’s apply our recognized methods, yielding the “improved” setup:

All strategies can now discover options considerably faster — simply Anticipated Sarsa falls out. This might very nicely be — it’s considerably much less used than Q-learning or Sarsa, and possibly extra a theoretical idea.

Thus, let’s proceed with out this technique and see how giant world sizes we are able to clear up:

Q-learning can now additionally clear up grid sizes of 25 x 25 with out issues — however Sarsa and Double Q-learning begin to degrade.

Extra particulars might be present in my introductory put up about TD strategies.

Dialogue

Within the improved setup, TD strategies usually carry out nicely. We solely eradicated Anticipated Sarsa early, which anyhow is just not such a typical algorithm.

“Easy” Sarsa and Double Q-learning battle for bigger atmosphere sizes, whereas Q-learning performs nicely total. The latter is considerably stunning, since Double Q-learning ought to deal with among the shortcomings of ordinary Q-learning, particularly the excessive variance. Doubtlessly, we already scale back the variance by operating every experiment n occasions. One other speculation might be that Double Q-learning takes longer to converge, for the reason that variety of parameters has additionally doubled — which might point out that the ability of Double Q-learning reveals higher for extra advanced issues with extra time.

As talked about performs Q-learning higher than Sarsa. This mirrors what can see in analysis / literature, particularly that Q-learning is considerably extra in style. This could most likely defined by it being off-policy, which normally yields extra highly effective resolution strategies. Sarsa however performs higher for stochastic or “harmful” duties: since in Sarsa the precise chosen motion is taken under consideration within the worth replace, it higher understands the results of its actions, which turns out to be useful for stochastic environments and / or environments the place one can, e.g., fall off a cliff. Regardless of the latter being the case right here, the atmosphere might be not advanced or giant sufficient, that this impact comes into play.

TD-n

TD-n strategies in a means marry classical TD studying and MC strategies. As Sutton so properly places it, they “free us from the tyranny of the timestep” [1]. In MC strategies, we’re compelled to attend a full episode earlier than making any updates. In TD strategies, we replace estimates in each step — however are additionally compelled to solely look one step sooner or later.

Thus, it is sensible to introduce n-step returns:

With that, we are able to merely introduce Sarsa-n:

We play episodes following the present coverage, after which replace the worth estimates with the n-step return.

In my corresponding put up, we additionally introduce an off-policy model of this. Nonetheless, to not blow up this put up too lengthy, and destructive expertise with off-policy MC strategies, we deal with the “classics” — equivalent to Sarsa-n — and tree-n tree backup, which we introduce subsequent.

n-step tree backup is an extension of the beforehand seen Anticipated Sarsa. When computing the n-step return, the corresponding transition tree seems to be as follows:

I.e., there’s a single path down the tree similar to the precise motion taken. Simply as in Anticipated Sarsa, we now wish to weigh actions based on their likelihood decided by the coverage. However since now now we have a tree of depth > 1, the cumulative worth of later ranges is weighted by the likelihood of the motion taken to succeed in these ranges:

The pseudocode seems to be as follows:

Right here’s my corresponding put up on n-step TD strategies.

Outcomes

As common, we begin with the “naïve” setting, and acquire the next outcomes:

Sarsa-n begins to battle already with smaller grid world sizes. Let’s see if the improved setup modifications this:

Now certainly Sarsa-n performs a lot better, however n-step tree backup doesn’t.

Dialogue

I discovered this discovery surprising and considerably arduous to clarify. I’d love to listen to your ideas on this — however within the meantime I used to be chatting with my chat agent of selection, and got here to this speculation: intermediate rewards doubtlessly confuse the tree algorithm, because it must study an identical return distribution over all potential actions. Additional, the extra ε decays, the extra the anticipated distribution would possibly differ from the habits coverage.

Mannequin-Primarily based Reinforcement Studying / Planning

Within the earlier chapter we mentioned the subject “planning” — within the RL context with this we primarily confer with model-based strategies. That’s: now we have (or construct) a mannequin of the atmosphere, and use this mannequin to discover additional “just about”, and particularly use these explorations for extra and higher updates / learnings of the worth operate. The next picture shows the combination of planning into studying very nicely:

Within the top-right nook we see the “classical” RL coaching loop (additionally dubbed “direct” RL): beginning with some worth operate / coverage we act within the (actual) atmosphere, and use this expertise to replace our price operate (or coverage within the case of policy-gradient strategies). When incorporating planning, we moreover additionally study a mannequin of the world from this expertise, after which use this mannequin to generate additional (digital) expertise, and replace our price or coverage operate from this.

This truly is precisely the Dyna-Q algorithm, which seems to be as follows in pseudocode:

Steps (a) — (d) are our classical Q-learning, whereas the remainder of the algorithm provides the novel planning performance, particularly the world mannequin studying.

One other associated algorithm is Prioritized Sweeping, which modifications how we pattern states for the “planning loop”: we discover and play in the true atmosphere, whereas studying the mannequin, and save state-action pairs with giant anticipated worth modifications to a queue. Solely with this queue we begin the “planning loop”, i.e. one thing to the steps (e) and (f) above:

Extra particulars might be present in my earlier put up on model-based RL strategies.

Outcomes

Let’s begin with the naïve setting:

Dyna Q performs moderately nicely, whereas Prioritized Sweeping struggles early on.

Within the improved setting we see an analogous factor:

Dialogue

Prioritized sweeping already carried out poorly within the corresponding introductory put up — I believe there both is a few concern, or extra possible this merely is a “tuning” factor — i.e. utilizing a fallacious sampling distribution.

Dyna-Q yields stable outcomes.

Benchmarking the Greatest Algorithms

We now have now seen the efficiency of all algorithms from Half I of Sutton’s guide by benchmarking them per chapter and on Gridworlds of as much as measurement 25 x 25. Already right here we noticed higher and worse performing algorithms, and particularly already discarded a number of candidates not suited to bigger environments.

Now we wish to benchmark the remaining ones — the most effective ones from every chapter — in opposition to each other, on Gridworlds as much as measurement 50 x 50.

These algorithms are:

worth iteration
on-policy MC
Q-learning
Sarsa-n
Dyna-Q

Outcomes

Right here’s how they carry out on Gridworld, this time with a maximal step restrict of 200.000:

Let’s additionally plot the corresponding time wanted (be aware that I plot unsuccessful runs — runs reaching the maximal variety of steps with out producing a possible coverage — at 500s):

We are able to observe a number of fascinating info from these figures:

The variety of steps vs. time wanted is very correlated.
Worth iteration performs exceptionally nicely, fixing even Gridworlds of measurement 50 x 50 with ease, and doing so magnitudes quicker than the next-best algorithm.
The rating for the remaining algorithms is (higher to worse): On-policy MC, Dyna-Q, Q-learning, Sarsa-n.

Within the subsequent part we focus on these in additional particulars.

Dialogue

1. Steps vs. Time

We began this put up with a dialogue on which metrics / measurement to make use of, and — particularly — whether or not to make use of variety of steps or time wanted to resolve the issue. Trying again, we are able to say that this dialogue was not so related in any case, and — considerably surprisingly — these two numbers are extremely correlated. That’s, even if, as initially described, one “step” can differ relying on the algorithm.

2. Worth Iteration Dominates

Worth Iteration carried out remarkably nicely, fixing even giant Gridworlds (as much as 50×50) with ease—outpacing all different algorithms by a large margin. This is likely to be stunning, contemplating that DP strategies are sometimes thought of theoretical instruments, not often utilized in observe. Actual-world purposes are likely to favor strategies like Q-learning [2], PPO [4], or MCTS [5].

So why does such a “textbook” technique dominate right here? As a result of this atmosphere is tailored for it:

The mannequin is totally recognized.
The dynamics are easy and deterministic.
The state area is comparatively small.

These are precisely the situations underneath which DP thrives. In distinction, model-free strategies like Q-learning are designed for settings the place such info is not out there. Their energy lies in generality and scalability, not in exploiting small, well-defined issues. Q-learning incurs excessive variance and requires many episodes to converge—disadvantages which can be magnified in small-scale environments. In brief, there’s a transparent trade-off between effectivity and generality. We’ll revisit this level in a future put up after we introduce operate approximation, the place Q-learning has extra room to shine.

3. A Rating Emerges

Past Worth Iteration, we noticed the next efficiency rating: On-policy MC > Dyna-Q > Q-learning > Sarsa-n

On-policy Monte Carlo emerged because the best-performing model-free algorithm. This suits with our earlier reasoning: MC strategies are easy, unbiased, and well-suited to issues with deterministic targets—particularly when episodes are comparatively brief. Whereas not scalable to giant or steady issues, MC strategies appear to be fairly efficient in small to medium-sized duties like Gridworld.

Dyna-Q comes subsequent. This end result reinforces our expectations: Dyna-Q blends model-based planning with model-free studying. Though the mannequin is discovered (not given, as in Worth Iteration), it’s nonetheless easy and deterministic right here—making the discovered mannequin helpful. This boosts efficiency considerably over pure model-free approaches.

Q-learning, whereas nonetheless highly effective, underperforms on this context for the explanations mentioned above: it’s a general-purpose algorithm that isn’t in a position to totally leverage the construction of straightforward environments.

Sarsa-n landed in final place. A possible clarification is the added bias launched by way of bootstrapping in its multi-step updates. In contrast to Monte Carlo strategies, which estimate returns from full trajectories (unbiased), Sarsa-n makes use of bootstrapped estimates of future rewards. In small environments, this bias can outweigh the advantages of decreased variance.

Lastly, let’s evaluate our outcomes vs. those from Sutton:

Observe that Sutton lists the overall variety of steps on the x-axis, whereas we checklist n, with the overall variety of states being n x n. For 376 states, Sutton report ~100k steps earlier than the optimum resolution is discovered, whereas we report 75k for 400 states (20 x 20), contemplating Dyna-Q. The numbers are extremely comparable and supply a reassuring validation of our setup and implementation.

Conclusion

This put up served each as a recap of our sequence on Half I of Sutton and Barto’s Reinforcement Studying [1]and as an extension past the guide’s scope—by benchmarking all launched algorithms on more and more bigger Gridworld environments.

We started by outlining our benchmarking setup, then revisited the core chapters of Half I: Dynamic Programming, Monte Carlo strategies, Temporal-Distinction studying, and Mannequin-Primarily based RL / Planning. In every part, we launched key algorithms equivalent to Q-learning, offered full Python implementations, and evaluated their efficiency on Gridworlds as much as measurement 25×25. The purpose of this preliminary spherical was to determine prime performers from every algorithmic household. Primarily based on our experiments, the standouts had been:
Worth Iteration, On-policy MC, Q-learning, Sarsa-n, and Dyna-Q. Python code to breed these outcomes, and particularly implementations of all mentioned strategies, is offered on GitHub.

Subsequent, we stress-tested these high-performers on bigger environments (as much as 50×50) and noticed the next rating:
Worth Iteration > On-policy MC > Dyna-Q > Q-learning > Sarsa-n

Whereas this end result could also be stunning—given the widespread use of Q-learning and the comparatively uncommon software of Worth Iteration and MC strategies—it is sensible in context. Easy, fully-known, deterministic environments are perfect for Worth Iteration and MC strategies. In distinction, Q-learning is designed for extra advanced, unknown, and high-variance environments the place operate approximation turns into crucial. As we mentioned, there’s a trade-off between effectivity in structured duties and generality in advanced ones.

That brings us to what’s subsequent. In upcoming posts, we’ll push the boundaries additional:

First, by benchmarking these strategies in tougher environments equivalent to two-player video games, the place direct competitors will expose their variations extra starkly.
Then, we’ll dive into Half II of Sutton’s guide, the place operate approximation is launched. This unlocks the power to scale reinforcement studying to environments far past what tabular strategies can deal with.

For those who’ve made it this far—thanks for studying! I hope you loved this deep dive, and I’d like to have you ever again for the subsequent installment within the sequence.