1. Introduction
Join optimization has been studied for more than four decades (selinger1979access, ) and continues to be an active area of research (trummer2017solving, ; neumann2018adaptive, ; marcus2018deep, ). The problem’s combinatorial complexity leads to the ubiquitous use of heuristics. For example, classical System Rstyle dynamic programs often restrict their search space to certain shapes (e.g., “leftdeep” plans). Query optimizers sometimes apply further heuristics to large join queries using genetic (postgresgenetic, ) or randomized (neumann2018adaptive, ) algorithms. In edge cases, these heuristics can break down (by definition), which results in poor plans (leis2015good, ).
In light of recent advances in machine learning, a new trend in database research explores replacing programmed heuristics with learned ones
(marcus2018towards, ; kipf2018learned, ; ortiz2018learning, ; marcus2018deep, ; btree, ; kraska2018case, ; mitzenmacher2018model, ; ma2018query, ). Inspired by these results, this paper explores the natural question of synthesizing datasetspecific join search strategies using learning. Assuming a given cost model and plan space, can we optimize the search over all possible join plans for a particular dataset? The hope is to learn tailored search strategies from the outcomes of previous planning instances that dramatically reduce search time for future planning.Our key insight is that join ordering has a deep algorithmic connection with Reinforcement Learning (RL) (sutton1998reinforcement, ). Join ordering’s sequential structure is the same problem structure that underpins RL. We exploit this algorithmic connection to embed RL deeply into a traditional query optimizer; anywhere an enumeration algorithm is used, a policy learned from an RL algorithm can just as easily be applied. This insight enables us to achieve two key benefits. First, we can seamlessly integrate our solution into many optimizers with the classical System R architecture. Second, we exploit the nested structure of the problem to dramatically reduce the training cost, as compared to previous proposals for a “learning optimizer”.
To better understand the connection with RL, consider the classical “bottomup” dynamic programming solution to join ordering. The principle of optimality leads to an algorithm that incrementally builds a plan from optimal subplans of size two, size three, and so on. Enumerated subplans are memoized in a lookup table, which is consulted to construct a sequence of 1step optimal decisions. Unfortunately, the space and time complexities of exact memoization can be prohibitive. Qlearning, an RL algorithm (sutton1998reinforcement, ), relaxes the requirement of exact memoization. Instead, it formulates optimal planning as a prediction problem: given the costs of previously enumerated subplans, which 1step decision is most likely optimal? RL views the classic dynamic programming lookup table as a model—a data structure that summarizes enumerated subplans and predicts the value of the next decision. In concrete terms, Qlearning sets up a regression from the decision to join a particular pair of relations to the observed benefit of making that join on past data (i.e., impact on the final cost of the entire query plan).
To validate this insight, we built an RLbased optimizer DQ that optimizes selectprojectjoin blocks and performs join ordering as well as physical operator selection. DQ observes the planning results of previously executed queries and trains an RL model to improve future search. We implement three versions of DQ to illustrate the ease of integration into existing DBMSes: (1) A standalone version built on top of Apache Calcite (calcite, ), (2) a version integrated with PostgreSQL (postgres, ), and (3) a version integrated with SparkSQL (armbrust2015spark, ). Deploying DQ into existing productiongrade systems (2) and (3) each required changes of less than 300 lines of code and training data could be collected through the normal operation of the DBMS with minimal overhead.
One might imagine that training such a model is extremely dataintensive. While RL algorithms are indeed notoriously datainefficient (typical RL settings, such as the Atari games (mnih2013playing, ), require hundreds of thousands of training examples), we can exploit the optimal subplan structure specific to join optimization to collect an abundance of highquality training data. From a single query that passes through a native optimizer, not only are the final plan and its total cost collected as a training example, so are all of its subplans and, recursively, everything inside the exact memoization table. For instance, planning an 18relation join query in TPCDS (Q64) through a bushy optimizer can yield up to 600,000 training data points thanks to DQ’s Qlearning formulation.
We thoroughly study this approach on two workloads: Join Order Benchmark (leis2015good, ) and TPCDS (tpcds, ).DQ sees significant speedups in planning times (up to ) relative to dynamic programming enumeration while essentially matching the execution times of optimal plans computed by the native enumerationbased optimizers. These planning speedups allow for broadening the plan space to include bushy plans and Cartesian products. In many cases, they lead to improved query execution times as well. DQ is particularly useful under nonlinear cost models such as memory limits or materialization. On two simulated cost models with significant nonlinearities, DQ improves on the plan quality of the next best heuristic over a set of 6 baselines by and . Thus, we show DQ approaches the optimization time efficiency of programmed heuristics and the plan quality of optimal enumeration.
We are enthusiastic about the general trend of integrating learning techniques into database systems—not simply by blackbox application of AI models to improve heuristics, but by the deep integration of algorithmic principles that span the two fields. Such an integration can facilitate new DBMS architectures that take advantage of all of the benefits of modern AI: learn from experience, adapt to new scenarios, and hedge against uncertainty. Our empirical results with DQ span across multiple systems, multiple cost models, and workloads. We show the benefits (and current limitations) of an RL approach to join ordering and physical operator selection. Understanding the relationships between RL and classical methods allowed us to achieve these results in a dataefficient way. We hope that DQ represents a step towards a future learning query optimizer.
2. Background
The classic join ordering problem is, of course, NPhard, and practical algorithms leverage heuristics to make the search for a good plan efficient. The design and implementation of optimizer search heuristics are wellunderstood when the cost model is roughly linear, i.e., the cost of a join is linear in the size of its input relations. This assumption underpins many classical techniques as well as recent work (selinger1979access, ; krishnamurthy1986optimization, ; trummer2017solving, ; neumann2018adaptive, ). However, many practical systems have relevant nonlinearities in join costs. For example, an intermediate result exceeding the available memory may trigger partitioning, or a relation may cross a size threshold that leads to a change in physical join implementation.
It is not difficult to construct reasonable scenarios where classical heuristics dramatically fail (Figure 1). Consider the query workload and dataset in the Join Order Benchmark (leis2015good, ). A popular heuristic from the original Selinger optimizer is to prune the search space to only include leftdeep join orders. Prior work showed that leftdeep plans are extremely effective on this benchmark for cost models that prefer index joins (leis2015good, ). Experimentally, we found this to be true as well: the worstcase cost over the entire workload is only 2x higher than the true optimum (for an exponentially smaller search space). However, when we simply change the cost model to be more nonlinear, consisting of (1) hybrid hash join operators that spill partitions to disk when data size exceeds available memory, or (2) hash join operators that can reuse previously built hash tables, suddenly the leftdeep heuristic is no longer a good idea—it is almost 50x more costly than the true optimum.
These results illustrate that in a practical sense, the search problem is unforgiving: various heuristics have different weak spots where they fail by orders of magnitude relative to optimal. For example, success on such atypical or nonlinear cost models may require searching over “bushy” plans, not just leftdeep ones. With new hardware innovations (arulraj2017build, ) and a move towards serverless RDBMS architectures (aurora, ), it is not unreasonable to expect a multitude of new query cost models that significantly differ from existing literature, which might require a complete redesign of standard pruning heuristics. Ideally, instead of a fixed heuristic, we would want a strategy to guide the search space in a more datadriven way—tailoring the search to a specific database instance, query workload, and observed join costs. This sets up the main premise of the paper: would it be possible to use datadriven machine learning methods to identify such a heuristic from data?
2.1. Example
We focus on the classical problem of searching for a query plan made up of binary join operators and unary selections, projections, and access methods. We will use the following database of three relations denoting employee salaries as a running example throughout the paper:
Consider the following join query:
There are many possible orderings to execute this query. For example, one could execute the example query as , or as .
2.2. Reinforcement Learning
Bellman’s “Principle of Optimality” and the characterization of dynamic programming is one of the most important results in computing (bellman2013dynamic, )
. In addition to forming the basis of relational query optimization, it has a deep connection to a class of stochastic processes called Markov Decision Processes (MDPs), which formalize a wide range of problems from path planning to scheduling. In an MDP model, an agent makes a sequence of decisions with the goal of optimizing a given objective (e.g., improve performance, accuracy). Each decision is dependent on the current state, and typically leads to a new state. The process is “Markovian” in the sense that the system’s current state completely determines its future progression. Formally, an MDP consists of a fivetuple:
where describes a set of states that the system can be in, describes the set of actions the agent can take,
describes a probability distribution over new states given a current state and action, and
defines a distribution of initial states. is the reward of taking action a in state s. The reward measures the performance of the agent. The objective of an MDP is to find a decision policy , a function that maps states to actions, with the maximum expected reward:subject to 
As with dynamic programming in combinatorial problems, most MDPs are difficult to solve exactly. Note that the greedy solution, eagerly maximizing the reward at each step, might be suboptimal in the long run. Generally, analytical solutions to such problems scale poorly in the time horizon.
Reinforcement learning (RL) is a class of stochastic optimization techniques for MDPs (sutton1998reinforcement, )
. An RL algorithm uses sampling, taking randomized sequences of decisions, to build a model that correlates decisions with improvements in the optimization objective (cumulative reward). The extent to which the model is allowed to extrapolate depends on how the model is parameterized. One can parameterize the model with a table (i.e., exact parameterization) or one can use any function approximator (e.g., linear functions, nearest neighbors, or neural networks). Using a neural network in conjunction with RL, or Deep RL, is the key technique behind recent results like learning how to autonomously play Atari games
(mnih2015human, ) and the game of Go (silver2016mastering, ).2.3. Markov Model of Enumeration
Now, we will review standard “bottomup” join enumeration, and then, we will make the connection to a Markov Decision Process. Every join query can be described as a query graph, where edges denote join conditions between tables and vertices denote tables. Any dynamic programming join optimizer implementation needs to keep track of its progress: what has already been done in a particular subplan (which relations were already joined up) and what options remain (which relations–whether base or the result of joins–can still be “joined in” with the subplan under consideration). The query graph formalism allows us to represent this state.
Definition 2.1 (Query Graph).
A query graph is an undirected graph, where each relation is a vertex and each join predicate defines an edge between vertices. Let denote the number of connected components of .
Making a decision to join two subplans corresponds to picking two vertices that are connected by an edge and merging them into a single vertex. Let be a query graph. Applying a join to the graph defines a new graph with the following properties: (1) and are removed from , (2) a new vertex is added to , and (3) the edges of are the union of the edges incident to and . Each join reduces the number of vertices by . Each plan can be described as a sequence of such joins until . The above description embraces another System R heuristic: “avoiding Cartesian products”. We can relax that heuristic by simply adding edges to at the start of the algorithm, to ensure it is fully connected.
Going back to our running example, suppose we start with a query graph consisting of the vertices . Let the first join be ; this leads to a query graph where the new vertices are . Applying the only remaining possible join, we arrive at a single remaining vertex corresponding to the join plan .
The join optimization problem is to find the best possible join sequence—i.e., the best query plan. Also note that this model can be simply extended to capture physical operator selection as well. The set of allowed joins can be typed with an eligible join type, e.g., or . We assume access to a cost model
, i.e., a function that estimates the incremental cost of a particular join.
Problem 1 (Join Optimization Problem).
Let define a query graph and define a cost model. Find a sequence terminating in to minimize:
subject to 
Note how this problem statement exactly defines an MDP (albeit by convention a minimization problem rather than maximization). is a representation of the state, is a representation of the action, the vertex merging process defines the state transition , and the reward function is the negative cost . The output of an MDP is a function that maps a given query graph to the best next join. Before proceeding, we summarize our notation in Table 1.
2.4. Long Term Reward of a Join
Symbol  Definition 

A query graph. This is a state in the MDP.  
A join. This is an action.  
The resultant query graph after applying a join.  
A cost model that scores joins. 
To introduce how RL gives us a new perspective on this classical database optimization problem, let us first examine the greedy solution. A naive solution is to optimize each independently (also called Greedy Operator Optimization (neumann2018adaptive, )). The algorithm proceeds as follows: (1) start with the query graph, (2) find the lowest cost join, (3) update the query graph and repeat until only one vertex is left.
The greedy algorithm, of course, does not consider how local decisions might affect future costs. For illustration, consider our running example query with the following simple costs (assume a single join method with symmetric cost):
The greedy solution would result in a cost of 140 (because it neglects the future effects of a decision), while the optimal solution has a cost of 110. However, there is an upside: this greedy algorithm has a computational complexity of , despite the superexponential search space.
The greedy solution is suboptimal because the decision at each index fails to consider the longterm value of its action. One might have to sacrifice a short term benefit for a long term payoff. Consider the optimization problem for a particular query graph :
(1) 
In classical treatments of dynamic programming, this function is termed the value function. It is noted that optimal behavior over an entire decision horizon implies optimal behavior from any starting index as well, which is the basis for the idea of dynamic programming. Conditioned on the current join, we can write in the following form:
leading to the following recursive definition of the Qfunction (or costtogo function):
(2) 
Intuitively, the Qfunction describes the longterm value of each join: the cumulative cost if we act optimally for all subsequent joins after the current join decision. Knowing is equivalent to solving the problem since local optimization is sufficient to derive an optimal sequence of join decisions.
If we revisit the greedy algorithm, and revise it hypothetically as follows: (1) start with the query graph, (2) find the lowest Qvalue join, (3) update the query graph and repeat, then this algorithm has the same computational complexity of but is provably optimal. To sketch out our solution, we will use Deep RL to approximate a global Qfunction (one that holds for all query graphs in a workload), which gives us a polynomialtime algorithm for join optimization.
2.5. Applying Reinforcement Learning
An important class of reinforcement learning algorithms, called Qlearning algorithms, allows us to approximate the Qfunction from samples of data (sutton1998reinforcement, ). What if we could regress from features of to the future cumulative cost based on a small number of observations? Practically, we can observe samples of decision sequences containing tuples, where is the query graph, is a particular join, is the cost of the join, and is the resultant graph. Such a sequence can be extracted from any final join plan and by evaluating the cost model on the subplans.
Let’s further assume we have a parameterized model for the Qfunction, :
where is a feature vector representing the query graph and
is a feature vector representing a particular join.
is the model parameters that represent this function and is randomly initialized at the start. For each training tuple , one can calculate the following label, or the “estimated” Qvalue:The can then be used as labels in a regression problem. If were the true Qfunction, then the following recurrence would hold:
So, the learning process, or Qlearning, defines a loss at each iteration:
Then parameters of the Qfunction can be optimized with gradient descent until convergence.
RL yields two key benefits: (1) the search cost for a single query relative to traditional query optimization is radically reduced, since the algorithm has the timecomplexity of greedy search, and (2) the parameterized model can potentially learn across queries that have “similar” but nonidentical subplans. This is because the similarity between subplans are determined by the query graph and join featurizations, and ; thus if they are designed in a sufficiently expressive way, then the neural network can be trained to extrapolate the Qfunction estimates to an entire workload.
The specific choice of Qlearning is important here (compared to other RL algorithms). First, it allows us to take advantage of optimal substructures during training and greatly reduce data needed. Second, compared to policy learning (marcus2018deep, ), Qlearning outputs a score for each join that appears in any subplan rather than simply selecting the best join. This is more amenable to deep integration with existing query optimizers, which have additional state like interesting orders and their own pruning of plans. Third, the scoring model allows for topk planning rather than just getting the best plan. We note that the design of Qlearning variants is an active area of research in AI (hester2017deep, ; van2016deep, ), so we opted for the simplicity of a Deep Qlearning approach and defer incorporation of advanced variants to future work.
2.6. Reinforcement Learning vs. Supervised Learning
Reinforcement Learning and Supervised Learning can seem very similar since the underlying inference methods in RL algorithms are often similar to those used in supervised learning and statistical estimation. Here is how we justify our terminology. In supervised learning, one has paired training examples with groundtruth labels (e.g., an image with a labeled object). For join optimization, this would mean a dataset where the example is the current join graph and the label is the next best join decision from an oracle. In the context of sequential planning, this problem setting is often called Imitation Learning
(osa2018algorithmic, ); where one imitates an oracle as best as possible.As in (levine2018learning, ), the term “Reinforcement Learning” refers to a class of empirical solutions to Markov Decision Process problems where we do not have the groundtruth, optimal next steps; instead, learning is guided by numeric “rewards” for next steps. In the context of join optimization, these rewards are subplan costs. RL rewards may be provided by a realworld experiment, a simulation model, or some other oracular process. In our work below, we explore different reward functions including both realworld feedback (§5) and simulation via traditional plan cost estimation (§3.3).
RL purists may argue that access to any optimization oracle moves our formulation closer to supervised learning than classical RL. We maintain this terminology because we see the pretraining procedure as a useful prior. Rather than expensive, ab initio learning from executions, we learn a useful (albeit imperfect) join optimization policy offline. This process bootstraps a more classical “learningbydoing” RL process online that avoids executing grossly suboptimal query plans.
There is additionally subtlety in the choice of algorithm. Most modern RL algorithms collect data episodically (execute an entire query plan and observe the final result). This makes sense in fields like robotics or autonomous driving where actions may not be reversible or decomposable. In query optimization, every query consists of subplans (each of which is its own “query”). Episodic data collection ignores this compositional structure.
3. Optimizer Architecture
Selinger’s optimizer design separated the problem of plan search from cost/selectivity estimation (selinger1979access, ). This insight allowed independent innovation on each topic over the years. In our initial work, we follow this lead, and intentionally focus on learning a search strategy only. Even within the search problem, we focus narrowly on the classical selectprojectjoin kernel. This too is traditional in the literature, going back to Selinger (selinger1979access, ) and continuing as recently as Neumann et al.’s very recent experimental work (neumann2018adaptive, ). It is also particularly natural for illustrating the connection between dynamic programming and Deep RL and implications for query optimization. We intend for our approach to plug directly into a Selingerbased optimizer architecture like that of PostgreSQL, DB2 and many other systems.
In terms of system architecture, DQ
can be simply integrated as a learningbased replacement for prior algorithms for searching a plan space. Like any nonexhaustive query optimization technique, our results are heuristic. The new concerns raised by our approach have to do with limitations of training, including overfitting and avoiding highvariance plans. We use this section to describe the extensibility of our approach and what design choices the user has at her disposal.
3.1. Overview
Now, we describe what kind of training data is necessary to learn a Qfunction. In supervised regression, we collect data of the form (feature, values). The learned function maps from feature to values. One can think of this as a stateless prediction, where the underlying prediction problem does not depend on some underlying process state. On the other hand, in the Qlearning setting, there is state. So we have to collect training data of the form (state, decision, new state, cost). Therefore, a training dataset has the following format (in Java notation):
In many cases like robotics or gameplaying, RL is used in a live setting where the model is trained onthefly based on concrete moves chosen by the policy and measured in practice. Qlearning is known as an “offpolicy” RL method. This means that its training is independent of the data collection process and can be suboptimal—as long as the training data sufficiently covers the decisions to be made.
3.2. Architecture and API
DQ collects training data sampled from a cost model and a native optimizer. It builds a model which improves future planning instances. DQ makes relatively minimal assumptions about the structure of the optimizer. Below are the API hooks that it requires implemented.
Workload Generation. A function that returns a list of training queries of interest. DQ requires a relevant workload for training. In our experiments, we show that this workload can be taken from query templates or sampled from the database schema.
Cost Sampling. A function that given a query returns a list of join actions and their resultant costs. DQ requires the system to have its own optimizer to generate training data. This means generating feasible join plans and their associated costs. Our experiments evaluate integration with deterministic enumeration, randomized, and heuristic algorithms.
Predicate Selectivity Estimation. A function that returns the selectivity of a particular single table predicate. DQ leverages the optimizer’s own selectivity estimate for featurization (§4.1).
In our evaluation (§6), we will vary these exposed hooks to experiment with different implementations for each (e.g., comparing training on highly relevant data from a desired workload vs. randomly sampling join queries directly from the schema).
3.3. Efficient Training Data Generation
Training data generation may seem onerous, but in fact, useful data is automatically generated as a consequence of running classical planning algorithms. For each join decision that the optimizer makes, we can get the incremental cost of the join. Suppose, we run a classical bushy dynamic programming algorithm to optimize a kway join, we not only get a final plan but also an optimal plan for every single subplan enumerated along the way. Each query generates an optimal query plan for all of the subplans that compose it, as well as observations of suboptimal plans that did not make the cut. This means that a single query generates a large amount of training examples. Figure 2 shows how the principle of optimality helps enhance a training dataset.
This data collection scheme differs from that of several popular RL algorithms such as PPO and Policy Gradients (schulman2017proximal, ) (and used in (marcus2018deep, )). These algorithms train their models “episodically”, where they apply an entire sequence of decisions and observe the final cumulative reward. An analogy would be a graph search algorithm that does not backtrack but resets to the starting node and tries the whole search again. While general, this scheme not suited for the structure of join optimization, where an optimal plan is composed of optimal substructures. Qlearning, an algorithm that does not rely on episodic data and can learn from offline data consisting of a hierarchy of optimal subplans, is a better fit for join optimization.



In our experiments, we bootstrap planning with a bushy dynamic program until the number of relations in the join exceeds 10 relations. Then, the data generation algorithm switches to a greedy scheme for efficiency for the last joins. Ironically, the data collected from such an optimizer might be “too good” (or too conservative) because it does not measure or learn from a diverse enough space of (costly, hence risky) subplans. If the training data only consisted of optimal subplans, then the learned Qfunction may not accurately learn the downside of poor subplans. Likewise, if purely random plans are sampled, the model might not see very many instances of good plans. To encourage more “exploration”, during data collection noise can be injected into the optimizer to force it to enumerate more diverse subplans. We control this via a parameter , the probability of picking a random join as opposed to a join with the lowest cost. As the algorithm enumerates subplans, if then a random (valid) join is chosen on the current query graph; otherwise it proceeds with the lowestcost join as usual. This is an established technique to address such “covariate shift”, a phenomenon extensively studied in prior work (laskey2017dart, ).
Query: <example query> AND Emp.id > 200 
Query: <example query> 
4. Realizing the Qlearning Model
Next, we present the mechanics of actually training and operating a Qlearning model.
4.1. Featurizing the Join Decision
Before we get into the details, we will give a brief motivation of how we should think about featurization in a problem like this. The features should be sufficiently rich that they capture all relevant information to predict the future cumulative cost of a join decision. This requires knowing what the overall query is requesting, the tables on the left side of the proposed join, and the tables on the right side of the proposed join. It also requires knowing how single table predicates affect cardinalities on either side of the join.
Participating Relations: The overall intuition is to use each column name as a feature, because it identifies the distribution of that column. The first step is to construct a set of features to represent which attributes are participating in the query and in the particular join. Let be the set of all attributes in the database (e.g., ). Each relation (including intermediate join results) has a set of visible attributes, , the attributes present in the output. Similarly, every query graph can be represented by its visible attributes . Each join is a tuple of two relations and we can get their visible attributes and . Each of the attribute sets can then be represented with a binary 1hot encoding: a value in a slot indicates that particular attribute is present, otherwise represents its absence. Using to denote concatenation, we obtain the query graph features, , and the join decision features, , and, finally, the overall featurization for a particular tuple is simply . Figure 3 illustrates the featurization of our example query.
Selections: Selections can change said distribution, i.e., (col, selpred) is different than (col, TRUE). To handle single table predicates in the query, we have to tweak the feature representation. As with most classical optimizers, we assume that the optimizer eagerly applies selections and projections to each relation. Next, we leverage the table statistics present in most RDBMS. For each selection in a query we can obtain the selectivity , which estimates the fraction of tuples present after applying the selection.^{1}^{1}1We consider selectivity estimation out of scope for this paper. See discussion in §3 and §7. To account for selections in featurization, we simply scale the slot in that the relation and attribute corresponds to, by . For instance, if selection is estimated to have a selectivity of , then the Emp.id slot in would be changed to . Figure 3(a) pictorially illustrates this scaling.
Physical Operators: The next piece is to featurize the choice of physical operator. This is straightforward: we add another onehot vector that indicates from a fixed set of implementations the type of join used (Figure 3(b)).
Extensibility: In this paper, we focus only on the basic form of featurization described above and study foreign key equality joins.^{2}^{2}2This is due to our evaluation workloads containing only such joins. §8 discusses how DQ could be applied to more general join types. An ablation study as part of our evaluation (Table 9) shows that the pieces we settled on all contribute to good performance. That said, there is no architectural limitation in DQ that prevents it from utilizing other features. Any property believed to be relevant to join cost prediction can be added to our featurization scheme. For example, we can add an additional binary vector to indicate which attributes have indexes built. Likewise, physical properties like sortorders can be handled by indicating which attributes are sorted in an operator’s output. Hardware environment variables (e.g., available memory) can be added as scalars if deemed as important factors in determining the final best plan. Lastly, more complex join conditions such as inequality conditions can also be handled (§8).
4.2. Model Training
DQ
uses a multilayer perceptron (MLP) neural network to represent the Qfunction. It takes as input the final featurization for a
pair, . Empirically, we found that a twolayer MLP offered the best performance under a modest training time constraint (minutes). The model is trained with a standard stochastic gradient descent (SGD) algorithm.
4.3. Execution after Training
After training, we obtain a parameterized estimate of the Qfunction, . For execution, we simply go back to the standard algorithm as in the greedy method but instead of using the local costs, we use the learned Qfunction: (1) start with the query graph, (2) featurize each join, (3) find the join with the lowest estimated Qvalue (i.e., output from the neural net), (4) update the query graph and repeat.
This algorithm has the timecomplexity of greedy enumeration except in greedy, the cost model is evaluated at each iteration, and in our method, a neural network is evaluated. One pleasant consequence is that DQ exploits the abundant vectorization opportunities in numerical computation. In each iteration, instead of invoking the neural net sequentially on each join’s feature vector, DQ batches all candidate joins (of this iteration) together, and invokes the neural net once on the batch. Modern CPUs, GPUs, and specialized accelerators (e.g., TPUs (jouppi2017datacenter, )) all offer optimized instructions for such singleinstruction multipledata (SIMD) workloads. The batching optimization amortizes each invocation’s fixed overheads and has the most impact on large joins.
5. Feedback From Execution
Optimizer  Cost Model 1  Cost Model 2  Cost Model 3  

Min  Mean  Max  Min  Mean  Max  Min  Mean  Max  
QuickPick (QP)  1.0  23.87  405.04  7.43  51.84  416.18  1.43  16.74  211.13 
IKKBZ (KBZ)  1.0  3.45  36.78  5.21  29.61  106.34  2.21  14.61  96.14 
Rightdeep (RD)  4.70  53.25  683.35  1.93  8.21  89.15  1.83  5.25  69.15 
Leftdeep (LD)  1.0  1.08  2.14  1.75  7.31  65.45  1.35  4.21  35.91 
Zigzag (ZZ)  1.0  1.07  1.87  1.0  5.07  43.16  1.0  3.41  23.13 
Exhaustive (EX)  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0 
DQ  1.0  1.32  3.11  1.0  1.68  11.64  1.0  1.91  13.14 
We have described how DQ learns from sampling the cost model native to a query optimizer. However, it is wellknown that a cost model (costs) may fail to correlate with reality (runtimes), due to poor cardinality estimates or unrealistic rules used in estimation. To correct these errors, the database community has seen proposals of leveraging feedback from execution (chaudhuri2008pay, ; markl2003leo, )
. We can perform an analogous operation on learned Qfunctions. Readers might be familiar with the concept of finetuning in the deep learning literature
(yosinski2014transferable, ), where a network is trained on one dataset and “transferred” to another with minimal retraining. DQ can optionally apply this technique to retrain itself on real execution runtimes to correlate better with the operating environment.5.1. Finetuning Dq
Finetuning DQ consists of two steps: pretraining as usual and retraining. First, DQ is pretrained to convergence on samples from the optimizer’s cost model; these are inexpensive to collect compared to real execution. Next, the weights of the first two layers of the neural network are frozen, and the output layer’s weights are reinitialized randomly. Retraining is then started on samples of real execution runtimes, which would only change the output layer’s weights.
Intuitively, the process can be thought of as first using the cost model to learn relevant features about the general structure of subplans (e.g., “which relations are generally beneficial to join?”). The retrained output layer then projects the effect of these features onto real runtimes. Due to its inexpensive nature, partial retraining is a common strategy applied in many machine learning applications.
5.2. Collecting Execution Data
For finetuning, we collect a list of realexecution data, (Graph, Join, Graph’, OpTime), where instead of the cost of the join, the real runtime attributed to the particular join operator is recorded. Peroperator runtimes can be collected by instrumenting the underlying system, or using the system’s native analysis functionality (e.g., EXPLAIN ANALYZE in Postgres).
6. Evaluation
We extensively evaluate DQ to investigate the following major questions:
To address the first two questions, we run experiments on standalone DQ. The last question is evaluated with endtoend experiments on DQintegrated Postgres and SparkSQL.
6.1. Standalone Optimization Experiments
We implemented DQ and a wide variety of optimizer search techniques previously benchmarked in Leis et al. (leis2015good, ) in a standalone Java query optimizer harness. Apache Calcite is used for parsing SQL and representing the SQL AST. We first evaluate standalone DQ and other optimizers for final plan costs; unless otherwise noted, exploration (§3.3) and realexecution feedback (§5) are turned off. We use the Join Order Benchmark (JOB) (leis2015good, ), which is derived from the real IMDB dataset (3.6GB in size; 21 tables). The largest table has 36 million rows. The benchmark contains 33 templates and 113 queries in total. The joins have between 4 and 15 relations, with an average of 8 relations per query.
We revisit a motivating claim from earlier: heuristics are wellunderstood when the cost model is linear but nonlinearities can lead to significant suboptimality. The experiments intend to illustrate that DQ offers a form of robustness to cost model, meaning, that it prioritizes plans tailored to the structure of the cost model, workload, and physical design—even when these plans are bushy.
We consider 3 cost models: CM1 is a model for a mainmemory database; CM2 additionally considers limited memory hash joins where after a threshold the costs of spilling partitions to disk are considered; CM3 additionally considers the reuse of alreadybuilt hash tables during upstream operators. We compare with the following baselines: QuickPick1000 (QP) (waas2000join, ) selects the best of 1000 random join plans; IKKBZ (KBZ) (krishnamurthy1986optimization, ) is a polynomialtime heuristic that decomposes the query graph into chains and orders them; dynamic programs Rightdeep (RD), Leftdeep (LD), Zigzag (ZZ) (ziane1993parallel, ), and Exhaustive (EX) exhaustively enumerate join plans with the indicated plan shapes. Details of the setup are listed in Appendix §A.
Results of this set of experiments are shown in Table 2.
6.1.1. Cost Model 1
Our results on CM1 reproduce the conclusions of Leis et al. (leis2015good, ), where leftdeep plans are generally good (utilize indexes well) and there is little need for zigzag or exhaustive enumeration. DQ is competitive with these optimal solutions without a priori knowledge of the index structure. In fact, DQ significantly outperforms the other heuristic solutions KBZ and QP. While it is true that KBZ also restricts its search to leftdeep plans, it is suboptimal for cyclic join graphs—its performance is hindered since almost all JOB queries contain cycles. We found that QP struggles with the physical operator selection, and a significant number of random samples are required to find a narrow set of good plans (ones the use indexes effectively).
Unsurprisingly, these results show that DQ, a learningbased solution, reasonably matches performance on cases where good heuristics exist. On average DQ is within 22% of the LD solution and in the worst case only 1.45 worse.
6.1.2. Cost Model 2
By simply changing to a different, yet realistic, cost model, we can force the leftdeep heuristics to perform poorly. CM2 accounts for disk usage in hybrid hash joins. In this cost model, none of the heuristics match the exhaustive search over the entire workload. Since the costs are largely symmetric for small relation sizes, there is little benefit to either leftdeep or rightdeep pruning. Similarly zigzag trees are only slightly better, and the heuristic methods fail by ordersofmagnitude on their worst queries.
DQ still comes close to the quality of exhaustive enumeration ( on average). It does not perform as well as in CM1 (with its worst query about the optimal cost) but is still significantly better than the alternatives. Results on CM2 suggest that as memory becomes more limited, heuristics begin to diverge more from the optimal solution. We explored this phenomenon further and report results in Table 3.
KBZ  1.0  3.31  30.64  41.64 
LD  1.0  1.09  6.45  6.72 
EX  1.0  1.0  1.0  1.0 
DQ  1.04  1.42  1.64  1.56 
6.1.3. Cost Model 3
Finally, we illustrate results on CM3 that allows for the reuse of hash tables. Rightdeep plans are no longer inefficient in this model as they facilitate reuse of the hash table (note right and left are simply conventions and there is nothing important about the labels). The challenge is that now plans have to contain a mix of leftdeep and rightdeep structures. Zigzag tree pruning heuristic was exactly designed for cases like this. Surprisingly, DQ is significantly ( on average and in the worst) better than zigzag enumeration. We observed that bushy plans were necessary in a small number of queries and DQ found such lowercost solutions.
In summary, results in Table 2 show that DQ is robust against different cost model regimes, since it learns to adapt to the workload at hand.
6.1.4. Planning Latency
Next, we report the planning (optimization) time of DQ and several other optimizers across the entire 113 JOB queries. The same model in DQ is used to plan all queries. Implementations are written in Java, singlethreaded^{3}^{3}3To ensure fairness, for DQ we configure the underlying linear algebra library to use 1 thread. No GPU is used., and reasonably optimized at the algorithmic level (e.g., QuickPick would shortcircuit a partial plan already estimated to be more costly than the current best plan)—but no significant efforts are spent on lowlevel engineering. Hence, the relative magnitudes are more meaningful than the absolute values. Experiments were run on an AWS EC2 c5.9xlarge instance with a 3.0GHz CPU and 72GB memory.
Figure 5 reports the runtimes grouped by number of relations. In the smalljoin regime, DQ’s overheads are attributed interfacing with a JVMbased deep learning library, DL4J (creating and filling the featurization buffers; JNI overheads due to native CPU backend execution). These could have been optimized away by targeting a nonJVM engine and/or GPUs, but we note that when the number of joins is small, exhaustive enumeration would be the ideal choice.
In the largejoin regime, DQ achieves drastic speedups: for the largest joins DQ runs up to 10,000 faster than exhaustive enumeration and than leftdeep. DQ upperbounds the number of neural net invocations by the number of relations in a query, and additionally benefits from the batching optimization (§4.3). We believe this is a profound performance argument for a learned optimizer—it would have an even more unfair advantage when applied to larger queries or executed on specialized accelerators (jouppi2017datacenter, ).
6.1.5. Quantity of Training Data
How much training data does DQ need to become effective? To study this, we vary the number of training queries given to DQ and plot the mean relative cost using the cross validation technique described before. Figure 6 shows the relationship. DQ requires about 6080 training queries to become competitive and about 30 queries to match the plan costs of QuickPick1000.
Digging deeper, we found that the breakeven point of 30 queries roughly corresponds to seeing all relations in the schema at least once. In fact, we can train DQ on small queries and test it on larger ones—as long as the relations are covered well. To investigate this generalization power, we trained DQ on all queries with 9 and 8 relations, respectively, and tested on the remaining queries (out of a total of 113). For comparison we include a baseline scheme of training on 80 random queries and testing on 33; see Table 4.
Table 4 shows that even when trained on subplans, DQ performs relatively well and generalizes to larger joins (recall, the workload contains up to 15way joins). This indicates that DQ indeed learns local structures—efficient joining of small combinations of relations. When those local structures do not sufficiently cover the cases of interest during deployment, we see degraded performance.
6.1.6. Relevance and Quality of Training Data
Quantity of training data matters, and so do relevance and quality. We first study relevance, i.e., the degree of similarity between the sampled training data and the test queries. This is controlled by changing the training data sampling scheme. Figure 7 plots the performance of different data sampling techniques each with 80 training queries. It confirms that the more relevant the training queries can be made towards the test workload, the less data is required for good performance.
# Training Queries  Mean Relative Cost  

Random  80  1.32 
Train 9way  82  1.61 
Train 8way  72  9.95 
Notably, it also shows that even synthetically generated random queries (R80) are useful. DQ still achieves a lower relative cost compared to QuickPick1000 even with random queries (4.16 vs. 23.87). This experiment illustrates that DQ does not actually require a priori knowledge of the workload.
Next, we study the quality of training data, i.e., the optimality of the native planner DQ observes and gathers data from. We collect a varying amount of data sampled from the native optimizer, which we choose to be QuickPick1000, leftdeep, or bushy (EX). Figure 8 shows that all methods allow DQ to quickly converge to good solutions. The DPbased methods, leftdeep and bushy, converge faster as they produce final plans and optimal subplans per query. In contrast, QuickPick yields only 1000 random full plans per query. The optimal subplans from the dynamic programs offer data variety valuable for training, and they cover better the space of different relation combinations that might be seen in testing.
6.2. Real Systems Execution
It is natural to ask: how difficult and effective is it for a productiongrade system to incorporate DQ? We address this question by integrating DQ into two systems, PostgreSQL and SparkSQL.^{4}^{4}4Versions: Spark 2.3; Postgres master branch checked out on 9/17/18. The integrations were found to be straightforward: Postgres and SparkSQL each took less than 300 LoC of changes; in total about two personweeks were spent.
6.2.1. Postgres Integration
DQ integrates seamlessly with the bottomup join ordering optimizer in Postgres. The original optimizer’s DP table lookup is replaced with the invocation of DQ
’ Tensorflow (TF) neural network through the TF C API. As discussed in §
6.1.4, plans are batchevaluated to amortize the TF invocation overhead. We run the Join Order Benchmark experiments on the integrated artifact and present the results below. All of the learning utilizes the cost model and cardinality estimates provided by Postgres.Training. DQ observes the native cost model and cardinality estimates from Postgres. We configured Postgres to consider bushy join plans (the default is to only consider leftdeep plans). These plans generate traces of joins and their estimated costs in the form described in §3.3. We do not apply any exploration and execute the native optimizer as is. Training data is collected via Postgres’ logging interface.
Table 5 shows that DQ can collect training data from an existing system with relatively minimal impact on its normal execution. The overhead can be further minimized if training data is asynchronously, rather than synchronously, logged.
Runtimes on JOB (Figure 9). We allow the Postgres query planner to plan over 80 of the 113 training queries. We use a 5fold cross validation scheme to hold out different sets of 33 queries. Therefore, each query has at least one validation set in which it was unseen during training. We report the worst case planning time and execution time for queries that have multiple such runs. In terms of optimization latency, DQ is significantly faster than Postgres for large joins, up to . For small joins there is a substantial overhead due to neural network evaluations (even though DQ needs score much fewer join orders). These results are consistent with the standalone experiment in Section 6.1.4 and the same comments there on smalljoin regimes apply. In terms of execution runtimes, DQ is significantly faster on a number of queries; averaging over the entire workload DQ yields a 14% speedup.
6.2.2. SparkSQL Integration
Median  Max  

Postgres, no collection  19.17 ms  149.53 ms 
Postgres, with collection  35.98 ms  184.22 ms 
DQ is also integrated into SparkSQL, a distributed data analytics engine. To show that DQ’s effectiveness applies to more than one workload, we evaluate the integrated result on TPCDS.
Training. SparkSQL 2.3 contains a costbased optimizer which enumerates bushy plans for queries whose number of relations falls under a tunable threshold. We set this threshold high enough so that all queries are handled by this bushy dynamic program. To score plans, the optimizer invokes DQ’s trained neural net through TensorFlow Java. We use the native SparkSQL cost model and cardinality estimates. All algorithmic aspects of training data collection remain the same as the Postgres integration.
Effectiveness on TPCDS (Figure 10). We collect data from and evaluate on 97 out of all 104 queries in TPCDS v2.4. The data files are generated with a scale factor of 1 and stored as columnar Parquet files. In terms of execution runtimes, DQ matches SparkSQL over the 97 queries (a mean speedup of 1.0). In terms of optimization runtimes, DQ has a mean speedup of 3.6 but a max speedup of 250 on the query with largest number of joins (Q64). Note that the mean optimization speedup here is less drastic than JOB because TPCDS queries contain much less relations to join.
Discussion. In summary, results above show that DQ’s effective not only on the one workload designed to stresstest joins, but also on a wellestablished decision support workload. Further, we demonstrate the ease of integration into productiongrade systems including a RDBMS and a distributed analytics engine. We hope these results provide motivation for developers of similar systems to incorporate DQ’s learningbased join optimization technique.
6.3. FineTuning With Feedback
Finally, we illustrate how DQ can overcome an inaccurate cost model by finetuning with feedback data (§5). We focus on a specific JOB query, Q10c, where the cost model particularly deviates from the true runtime. Baseline DQ is trained on data collected over 112 queries, which is every query except for Q10c, as usual (i.e., values are costs from Postgres’ native cost model). For finetuning we execute a varying amount of these queries and collect their actual runtimes. To encourage observing a variety of physical operators, we use an exploration parameter of when observing runtimes (recall from §3.3 exploration means with probability we form a random intermediate join).
Figure 11 shows the results as a function of the number of queries observed for real execution. Postgres emits a plan that executes in s, while baseline DQ emits a plan that executes in s. After finetuning, DQ emits a plan that executes in s, outperforming both Postgres and its original performance. This shows true runtimes are useful in correcting faulty cost model and/or cardinality estimates. Interestingly, training a version of DQ using only real runtimes failed to converge to a reasonable model—this suggests learning highlevel features from inexpensive samples from the cost model is beneficial.
7. Related Work
Application of machine learning in database internals is still the subject of significant debate this year and will continue to be a contentious question for years to come (btree, ; kraska2018case, ; mitzenmacher2018model, ; ma2018query, ). An important question is what problems are amenable to machine learning solutions. We believe that query optimization is one such subarea. The problems considered are generally hard and ordersofmagnitude of performance are at stake. In this setting, poor learning solutions will lead to slow but not incorrect execution, so correctness is not a concern.
Cost Function Learning We are certainly not the first to consider “learning” in the query optimizer and there are a number of alternative architectures that one may consider. The precursors to this work are attempts to correct query optimizers through execution feedback. One of the seminal works in this area is the LEO optimizer (markl2003leo, ). This optimizer uses feedback from the execution of queries to correct inaccuracies in its cost model. The underlying cost model is based on histograms. The basic idea inspired several other important works such as (chaudhuri2008pay, ). The sentiment in this research still holds true today; when Leis et al. extensively evaluated the efficacy of different query optimization strategies they noted that feedback and cost estimation errors are still challenges in query optimizers (leis2015good, ). A natural first place to include machine learning would be what we call Cost Function Learning, where statistical learning techniques are used to correct or replace existing cost models. This is very related to the problem of performance estimation of queries (akdere2012learning, ; wu2013predicting, ; wu2013towards, ).
We actually investigated this by training a neural network to predict the selectivity of a single relation predicate. Results were successful, albeit very expensive from a data perspective. To estimate selectivity on an attribute with 10k distinct values, the training set had to include 1000 queries. This architecture suffers from the problem of featurization of literals; the results are heavily dependent on learning structure in literal values from the database that are not always straightforward to featurize. This can be especially challenging for strings or other nonnumerical data types. A recent workshop paper does show some promising results in using Deep RL to construct a good feature representation of subqueries but it still requires 10k queries to train (ortiz2018learning, ).
Learning in Query Optimization Recently, there has been several exciting proposals in putting learning inside a query optimizer. Ortiz et al. (ortiz2018learning, ) applies deep RL to learn a representation of queries, which can then be used in downstream query optimization tasks. Liu (liu2015cardinality, ) and Kipf (kipf2018learned, ) use DNNs to learn cardinality estimates. Closer to our work is Marcus et al.’s proposal of a deep RLbased join optimizer, ReJOIN (marcus2018deep, ), which offered a preliminary view of the potential for deep RL in this context. The early results reported in (marcus2018deep, ) top out at a 20% improvement in plan execution time of Postgres (compared to our 3x), and as of that paper they had only evaluated on 10 out of the 113 JOB queries that we study here. DQ qualitatively goes beyond that work by offering an extensible featurization scheme supporting physical join selection. More fundamentally, DQ integrates the dynamic programming of Qlearning into that of a standard query optimizer, which allows us to use offpolicy learning. Due to use of onpolicy policy gradient methods, (marcus2018deep, ) requires about 8,000 training queries to reach native Postgres’ cost on the 10 JOB queries. DQ exploits optimal substructures of the problem and uses offpolicy Qlearning to increase dataefficiency by two orders of magnitude: 80 training queries to outperform Postgres’ real execution runtimes on the entire JOB benchmark.
Adaptive Query Optimization Adaptive query processing (avnur2000eddies, ; deshpande2007adaptive, ) as well as the related techniques to reoptimize queries during execution (markl2004robust, ; babu2005proactive, ) is another line of work that we think is relevant to the discussion. Reinforcement learning studies sequential problems and adaptive query optimization is a sequential decision problem over tuples rather than subplans. We focus our study on optimization in fixed databases and the adaptivity that DQ offers is at a workload level. Continuously updating a neural network can be challenging for very finegrained adaptivity, e.g., processing different tuples in different ways.
Robustness There are a couple of branches of work that study robustness to different parameters in query optimization. In particular, the field of “parametric query optimization” (hulgeri2002parametric, ; trummer2014multi, ), studies the optimization of piecewise linear cost models. Interestingly, DQ is it is agnostic to this structure. It learns a heuristic from data identifying different regimes where different classes of plans work. We hope to continue experiments and attempt to interpret how DQ is partitioning the feature space into decisions. There is also a deep link between this work and least expected cost (LEC) query optimization (chu2002least, ). Markov Decision Processes (the main abstraction in RL) are by definition stochastic and optimize the LEC objective.
Join Optimization At Scale Scaling up join optimization has been an important problem for several decades, most recently (neumann2018adaptive, ). At scale, several randomized approaches can be applied. There is a long history of randomized algorithms (e.g., the QuickPick algorithm (waas2000join, )
) and genetic algorithms
(bennett1991genetic, ; steinbrunn1997heuristic, ). These algorithms are pragmatic and it is often the case that commercial optimizers will leverage such a method after the number of tables grows beyond a certain point. The challenge with these methods is that their efficacy is hard to judge. We found that QuickPick often varied in performance on the same query quite dramatically.Another heuristic approach is relaxation, or solving the problem exactly under simplified assumptions. One straightforward approach is to simply consider greedy search avoiding Cartesian products (fegaras1998new, ), which is also the premise of the IKKBZ algorithms (ibaraki1984optimal, ; krishnamurthy1986optimization, ). Similar linearization arguments were also made in recent work (trummer2017solving, ; neumann2018adaptive, ). Existing heuristics do not handle all types of nonlinearities well, and this is exactly the situation where learning can help. Interestingly enough, our proposed technique has a runtime, which is similar to the linearizedDP algorithm described in (neumann2018adaptive, ). We hope to explore the very large join regime in the future and an interesting direction is to compare DQ to recently proposed techniques like (neumann2018adaptive, ).
8. Discussion, Limitations, and Conclusion
We presented our method with a featurization designed for inner joins over foreign key relations as these were the major join queries in our benchmarks. This is not a fundamental restriction and is designed to ease exposition. It is relatively straightforward to extend this model to join conditions composed of conjunctions of binary expressions. Assume the maximum number of expressions in the conjunction is capped at . As before, let be the set of all attributes in the database. Each expression has two attributes and an operator. As with featurizing the vertices we can 1hot encode the attributes present. We additionally have to 1hot encode the binary operators . For each of the expressions in the conjunctive predicate, we concatenate the binary feature vectors that have its operator and attributes. Since the maximum number of expressions in the conjunction capped at , we can get a fixed sized feature vector for all predicates.
More broadly, we believe DQ is a step towards a learning query optimizer. As illustrated by the Cascades optimizer (graefe1995cascades, ) and followon work, costbased dynamic programming—whether bottom up or topdown with memoization—needs not be restricted to selectprojectjoin blocks. Most query optimizations can be recast into a space of algebraic transformations amenable to dynamic programming, including asymmetric operators like outer joins, crossblock optimizations including order optimizations and “sideways information passing”, and even nonrelational operators like PIVOT. The connection between RL and Dynamic Programming presented in this paper can be easily leveraged in those scenarios as well. Of course this blows up the search space, and large spaces are ideal for solutions like the one we proposed.
It is popular in recent AI research to try “endtoend” learning, where problems that were traditionally factored into subproblems (e.g., selfdriving cars involve separate models for localization, obstacle detection and lanefollowing) are learned in a single unified model. One can imagine a similar architectural ambition for an endtoend learning query optimizer, which simply maps subplan features to measured runtimes. This would require a significant corpus of runtime data to learn from, and changes to the featurization and perhaps the deep network structure we used here. DQ is a pragmatic middle ground that exploits the structure of the join optimization problem. Further exploring the extremes of learning and query optimization in future work may shed more insights.
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Appendix A Standalone Optimization Experiment Setup
We consider three different cost models on the same workload:
CM1: In the first cost model (inspired by (leis2015good, )), we model a mainmemory database that performs two types of joins: index joins and inmemory hash joins. Let describe the current operator, be the left child operator, and be the right child operator. The costs are defined with the following recursions:
where c denotes the cost estimation function, is the cardinality function, and match denotes the expected cost of an index match, i.e., fraction of records that match the index lookup (always greater than 1) multiplied by a constant factor (we chose ). We assume indexes on the primary keys. In this cost model, if an eligible index exists it is generally desirable to use it, since rarely exceeds for foreign key joins. Even though the cost model is nominally “nonlinear”, primary tradeoff between the index join and hash join is due to index eligibility and not dependent on properties of the intermediate results. For the JOB workload, unless is set to be very high, hash joins have rare occurrences compared to index joins.
CM2: In the next cost model, we remove index eligibility from consideration and consider only hash joins and nested loop joins with a memory limit . The model charges a cost when data requires additional partitioning, and further falls back to a nested loop join when the smallest table exceeds the squared memory:
The nonlinearities in this model are sizedependent, so controlling the size of intermediate relations is important in the optimization problem. We set the memory limit to tuples in our experiments. This limit is low in realworld terms due to the small size of the benchmark data. However, we intend for the results to be illustrative of what happens in the optimization problems.
CM3: In the next cost model, we model a database that accounts for the reuse of alreadybuilt hash tables. We use the Gamma database convention where the left operator as the “build” operator and the right operator as the “probe” operator (gerber1986data, ). If the previous join has already built a hash table on an attribute of interest, then the hash join does not incur another cost.
We also allow for index joins as in CM1. This model makes hash joins substantially cheaper in cases where reuse is possible. This model favors some subplans to be rightdeep plans which maximize the reuse of the built hash tables. Therefore, optimal solutions have both leftdeep and rightdeep segments.
In our implementation of these cost models, we use true cardinalities on singletable predicates, and we leverage standard independence assumptions to construct more complicated cardinality estimates. (This is not a fundamental limitation of DQ. Results in §6.2 have shown that when Postgres and SparkSQL provide their native cost model and cardinality estimates, DQ is as effective.) The goal of this work is to evaluate the join ordering process independent of the strength or weakness of the underlying cardinality estimation.
We consider the following baseline algorithms. These algorithms are not meant to be a comprehensive list of heuristics but rather representative of a class of solutions.
Optimizer  Cost Model 1  Cost Model 2  Cost Model 3  

Min  Mean  Max  Min  Mean  Max  Min  Mean  Max  
QuickPick (QP)  1  23.87  405.04  7.43  51.84  416.18  1.43  16.74  211.13 
IKKBZ (KBZ)  1  3.45  36.78  5.21  29.61  106.34  2.21  14.61  96.14 
Rightdeep (RD)  4.7  53.25  683.35  1.93  8.21  89.15  1.83  5.25  69.15 
Leftdeep (LD)  1  1.08  2.14  1.75  7.31  65.45  1.35  4.21  35.91 
Zigzag (ZZ)  1  1.07  1.87  1  5.07  43.16  1  3.41  23.13 
Exhaustive (EX)  1  1  1  1  1  1  1  1  1 
DQ  1  1.32  3.11  1  1.68  11.64  1  1.91  13.14 
Minimum Selectivity (MinSel)  2.43  59.86  1083.12  23.46  208.23  889.7  9.81  611.1  2049.13 
IKKBZ+DP (LDP)  1  1.09  2.72  2.1  10.03  105.32  2.01  3.99  32.19 

[noitemsep]

Exhaustive (EX): This is a dynamic program that exhaustively enumerates all join plans avoiding Cartesian products.

leftdeep (LD): This is a dynamic program that exhaustively enumerates all leftdeep join plans.

RightDeep (RD): This is a dynamic program that exhaustively enumerates all rightdeep join plans.

ZigZag (ZZ): This is a dynamic program that exhaustively enumerates all zigzag trees (every join has at least one base relation, either on the left or the right) (ziane1993parallel, ).

IKKBZ (KBZ): This algorithm is a polynomial time algorithm that decomposes the query graph into chains and orders the chains based on a linear approximation of the cost model (krishnamurthy1986optimization, ).

QuickPick1000 (QP): This algorithm randomly selects 1000 join plans and returns the best of them. 1000 was selected to be roughly equivalent to the planning latency of DQ (waas2000join, ).

Minimum Selectivity (MinSel): This algorithm selects the join ordering based on the minimum selectivity heuristic (neumann2018adaptive, ). While MinSel was fast, we found poor performance on the 3 cost models used in the paper.

Linearized Dynamic Program (LDP): This approach applies a dynamic program in the innerloop of IKKBZ (neumann2018adaptive, ). Not surprisingly, LDP’s results were highly correlated with those of IKKBZ and LeftDeep enumeration, so we chose to omit them from the main body of the paper.
All of the algorithms consider join ordering without Cartesian products, so EX is an optimal baseline. We report results in terms of the suboptimality w.r.t. EX, namely . We present results on all 113 JOB queries. We train on 80 queries and test on 33 queries. We do 4fold cross validation to ensure that every test query is excluded from the training set at least once. The performance of DQ is only evaluated on queries not seen in the training workload.
Our standalone experiments are integrated with Apache Calcite (calcite, ). Apache Calcite provides libraries for parsing SQL, representing relational algebraic expressions, and a Volcanobased query optimizer (graefe1991volcano, ; graefe1995cascades, ). Calcite does not handle physical execution or storage and uses JDBC connectors to a variety of database engines and file formats. We implemented a package inside Calcite that allowed us to leverage its parsing and plan representation, but also augment it with more sophisticated cost models and optimization algorithms. Standalone DQ is written in singlethreaded Java. The extended results including omitted techniques are described in Table 6.
Appendix B Cost Model
We additionally omitted experiments with a simplified cost model only searching for join orders and ignoring physical operator selection. We fed in true cardinalities to estimate the selectivity of each of the joins, which is a perfect version of the “” model. We omitted these results as we did not see differences between the techniques and the goal of the study was to understand the performance of DQ over cost models that cause the heuristics to fail. In particular, we found that threshold nonlinearities as in CM3 cause the most problems.
Mean  

QP  1.02 
IKKBZ  1.34 
LD  1.02 
ZZ  1.02 
Ex  1 
DQ  1.03 
MinSel  1.11 
Appendix C Additional Standalone Experiments
In the subsequent experiments, we try to characterize when DQ is expected to work and how efficiently.
c.1. Sensitivity to Training Data
Classically, join optimization algorithms have been deterministic. Except for QP, all of our baselines are deterministic as well. Randomness in DQ (besides floatingpoint computations) stems from what training data is seen. We run an experiment where we provide DQ with 5 different training datasets and evaluate on a set of 20 holdout queries. We report the max range (worst factor over optimal minus best factor over optimal) in performance over all 20 queries in Table 7. For comparison, we do the same with QP over 5 trials (with a different random seed each time).
CM1  CM2  CM3  

QP  2.11  1.71  3.44 
DQ  1.59  1.13  2.01 
We found that while the performance of DQ does vary due to training data, the variance is relatively low. Even if we were to account for this worst case, DQ would still be competitive in our macrobenchmarks. It is also substantially lower than that of QP, a true randomized algorithm.
c.2. Sensitivity to Faulty Cardinalities
In general, the cardinality/selectivity estimates computed by the underlying RDBMS do not have uptodate accuracy. All query optimizers, to varying degrees, are exposed to this issue since using faulty estimates during optimization may yield plans that are in fact suboptimal. It is therefore worthwhile to investigate this sensitivity and try to answer, “is the neural network more or less sensitive than classical dynamic programs and heuristics?”
In this microbenchmark, the optimizers are fed perturbed base relation cardinalities (explained below) during optimization; after the optimized plans are produced, they are scored by an oracle cost model. This means, in particular, DQ only sees noisy relation cardinalities during training and is tested on true cardinalities. The workload consists of 20 queries randomly chosen out of all JOB queries; the join sizes range from 6 to 11 relations. The final costs reported below are the average from 4fold cross validation.
The perturbation of base relation cardinalities works as follows. We pick random relations, the true cardinality of each is multiplied by a factor drawn uniformly from . As increases, the estimate noisiness increases (errors in the leaf operators get propagated upstream in a compounding fashion). Table 8 reports the final costs with respect to estimate noisiness.
KBZ  6.33  6.35  6.35  5.85 

LD  5.51  5.53  5.53  5.60 
EX  5.51  5.53  5.53  5.60 
DQ  5.68  5.70  5.96  5.68 
Observe that, despite a slight degradation in the execution, DQ is not any more sensitive than the KBZ heuristic. It closely imitates exhaustive enumeration—an expected behavior since its training data comes from EX’s plans computed with the faulty estimates.
c.3. Ablation Study
Graph Features  Sel. Scaling  Loss  

No Predicates  No  No  0.087 
Yes  No  0.049  
Yes  Yes  0.049  
Predicates  No  No  0.071 
Yes  No  0.051  
Yes  Yes  0.020 
Appendix D Discussion about Postgres Experiment
We also run a version of DQ where the model is only trained with online data (effectively the setting considered in ReJOIN (marcus2018deep, )). Even on an idealized workload of optimizing a single query (Query 10c), we could not get that approach to converge. We believe that the discrepancy from (marcus2018deep, ) is due to physical operator selection. In that work, the Postgres optimizer selects the physical operators given the appropriate logical plans selected by the RL policy. With physical operator selection, the learning problem becomes significantly harder (Figure 12).
We initially hypothesized the DQ outperforms the native Postgres optimizer in terms of execution times since it considers bushy plans. This hypothesis only partially explains the results. We run the same experiment where DQ is restricted to producing leftdeep plans; in other words, DQ considers the same plan space as the native Postgres optimizer. We found that there was still a statistically significant speedup:
Mean  Max  

DQ:LD  1.09  2.68 
DQ:EX  1.14  2.72 
We speculate that the speedup is caused by imprecision in the Postgres cost model. As a learning technique, DQ may smooth out inconsistencies in the cost model.
Finally, we compare with Postgres’ genetic optimizer (GEQ) on the 10 largest joins in JOB. DQ is about 7% slower in planning time, but nearly 10
faster in execution time. The difference in execution is mostly due to one outlier query on which GEQ is 37
slower.