| Question | Context | Answer | Question | Context | Answer |
| What is a major importance of Southern California in relation to California and the US? | ...Southern California is a major economic center for the state of California and the US... | major economic center | What has something experienced? | Areas of the Baltic that have experienced eutrophication. | eutrophication |
| What is the translation from English to German? | Most of the planet is ocean water. | Der Groβteil der Erde ist Meerwasser | Who is the illustrator of Cycle of the Werewolf? | Cycle of the Werewolf is a short novel by Stephen King, featuring illustrations by comic book artist | Bernie Wrightson |
| What is the summary? | Harry Potter star Daniel Radcliffe gains access to a reported £320 million fortune... | Harry Potter star Daniel Radcliffe gets | What is the change in dialogue state? | Bernie Wrightson. Are there any Eritrean restaurants in town? | food: Eritrean |
| Hypothesis: Product and geography Premise: Conceptually cream are what make cream skimming | skimming has two basic | | What is the translation | The table has column names... Tell me what the notes | SELECT notes from table WHERE |
| work. Entailment, neutral, or contradiction? | dimensions - product and geography. | | from English to SQL? | are for South Australia | 'Current Slogan' = 'South Australia' |
| | | | | |
| | | | | |
| A stirring, funny and finally | | | | |
| | | | Joan made sure to thank | |
| transporting re-imagining of | | | | |
| | | Who had given help? | | |
| Is this sentence positive or negative? | Beauty and the Beast and 1930s horror film. | positive | Susan or Joan? | Susan for all the help | Susan |
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+ "html": "| Algorithm A.1 Pseudocode for the SAVE algorithm. |
| 1: procedure SAVE(θ) |
| 2: | while true do |
| 3: | Begin episode at s |
| 4: | while acting do |
| 5: | Estimate QmCTs(s,:) ← MCTS(s, Qθ) |
| 6: | Select a using epsilon-greedy from QMCTs(s,:) |
| 7: 8: | Execute a in environment and receive s',r |
| 9: | Add (s,a,r,s',QmCTs(s,·)) to replay buffer |
| s↑s` |
| 10: | while learning do |
| 11: | Sample minibatch of experience from the replay buffer |
| 12: | Update θ to minimize Equation 6 |
| 13: |
| 14: | procedure MCTS(so, Qθ) |
| 15: | Qo(s,a)←Qe(s,a) forall s,a |
| 16: | No(s,a) ←1for all s,a |
| 17: | k←0 |
| 18: 19: | while search budget remains (k < K) do |
| 20: | Traverse the search tree with πk (Equation 1) |
| 21: | Expand new state sT and add it to the search tree |
| 22: | Evaluate maxa Qe(sT,a) and backup returns (Equation 3) |
| 23: | Set Nk+1(s,a) ← Nk(s,a) and then increment counts of visited states and actions |
| Compute estimates for Qk+1(s,a) (Equation 4) |
| 24: | k←k+1 |
| 25: | Return {Qk(so,ai)}i |
+
+The SAVE agent is implemented as described in Section 3 and Algorithm A.1 provides additional pseudocode explaining the algorithm. In Algorithm A.1, we provide an example of using SAVE in an episode setting where learning happens after every episode; however, SAVE can be used in any Q-learning setup including in distributed setups where separate processes are concurrently acting and learning. In particular, in our experiments we use the distributed setup described in Section A.1. Note that when performing epsilon-greedy exploration (Line 6 of Algorithm A.1), we either choose an action uniformly at random with probability $\epsilon$ , and otherwise choose the action with the highest value of $Q _ { \mathrm { M C T S } }$ out of the actions which were explored during search (i.e., we do not consider actions that were not explored, even if they have a higher $Q _ { \mathrm { M C T S } } )$ . In all experiments (except tabular Tightrope), we use a UTC exploration constant of $c = 2$ , though we have found SAVE’s performance to be relatively robust to this parameter setting.
+
+# A.3 PUCT
+
+The PUCT search policy is based on that described by Silver et al. (2017a) and Silver et al. (2018). Specifically, we choose actions during search according to Equation 1, with:
+
+$$
+\begin{array} { c } { { Q _ { k } = \displaystyle \frac { \sum _ { i = 1 } ^ { N _ { k } ( s , a ) } R _ { i } ( s , a ) } { N _ { k } ( s , a ) } } } \\ { { { } } } \\ { { U _ { k } ( s , a ) = c \cdot \pi ( s , a ) \displaystyle \frac { \sqrt { \sum _ { a } N _ { k } ( s , a ) } } { N _ { k } ( s , a ) + 1 } } } \end{array}
+$$
+
+where $c$ is an exploration constant, $\pi ( s , a )$ is the prior policy, and $N _ { k } ( s , a )$ is the total number of times action $a$ had been taken from state $s$ at iteration $k$ of the search. Like Silver et al. (2017a; 2018), we add Dirichlet noise to the prior policy:
+
+$$
+\pi ( s , a ) = ( 1 - \epsilon ) \cdot \pi _ { \theta } ( s , a ) + \epsilon \eta ,
+$$
+
+where $\eta \sim \mathrm { D i r } ( 1 / n _ { \mathrm { a c t i o n s } } )$ . In our experiments we set $\epsilon = 0 . 2 5$ and $c = 2$ . During training, after search is complete, we sample an action to execute in the environment from $\pi _ { \mathrm { M C T S } } ( s _ { 0 } , a ) =$ $\begin{array} { r } { N _ { K } ( s _ { 0 } , a ) / \sum _ { a } { N _ { K } ^ { - } ( s _ { 0 } , a ) } } \end{array}$ . At test time, we select the action which has the maximum visit count (with random tie-breaking).
+
+To train the PUCT agent, we used separate policy $\pi _ { \theta } ( s , a )$ and value $V _ { \theta } ( s )$ heads which were trained using a combined loss (Equation 6), with:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { Q } = \frac { 1 } { N } \sum _ { \mathcal { D } } \big \| V _ { \boldsymbol { \theta } } ( s ) - R \big \| _ { 2 } } \\ { \displaystyle \mathcal { L } _ { A } = - \frac { 1 } { N } \sum _ { \mathcal { D } } \pi _ { \mathrm { M C T S } } ( s , \cdot ) ^ { \top } \log \pi _ { \boldsymbol { \theta } } ( s , \cdot ) } \end{array}
+$$
+
+where $R$ is the Monte-Carlo return observed from state $s$ . We used fixed values of $\beta _ { Q } = 0 . 5$ and $\beta _ { A } = 0 . 5$ in all our experiments with PUCT. We used the same replay and training setup as used in the Q-learning and SAVE agents, with two exceptions. First, we additionally include episodic Monte-Carlo returns $R$ and policies $\pi _ { \mathrm { M C T S } }$ in the replay buffer so they can be used during learning. Second, we did not use $\epsilon$ -greedy exploration (because the Dirichlet noise in the PUCT term already enables sufficient exploration).
+
+We tried several different hyperparameter settings and variants of the PUCT agent to attempt to improve the results. For example, we tried using a 1-step TD error for learning the values, which should have lower variance and thus result in more stable learning of values. We also tried reducing the replay ratio to 1 and the replay size to 400 in order to make the experience for training more on-policy. However, we did not find that these changes improved the results. We also tried different settings of $\epsilon$ for the Dirichlet noise, but found that lower values resulted in too little exploration, while higher values resulted in too much exploration.
+
+# B DETAILS ON TIGHTROPE
+
+# B.1 ENVIRONMENT
+
+The Tightrope environment has 11 states which are connected together in a chain. Each state has 100 actions, $M \%$ of which will cause the episode to terminate when executed and the rest of which will cause the environment to transition to the next state. Each state is represented using a vector of 50 random values drawn from a standard normal distribution, which are the same across episodes. The indices of terminal actions are selected randomly and are different for each state but are consistent across episodes. Agents always begin in the first state of the chain.
+
+In the sparse reward setting, we randomly select one of the states in the chain to be the “final” state (excluding the first state), to enable the agent to sometimes train on easy problems and sometimes train on hard problems. If the agent reaches this final state, it receives a reward of 1 and the episode terminates. If it takes a non-terminal action, it transitions to the next state in the chain and receives a reward of 0. Otherwise, if it takes a terminal action, the episode terminates and the agent receives a reward of 0.
+
+In the dense reward setting, the “final” state is always chosen to be the last state in the chain. If the agent reaches the final state in the chain, it receives a reward of 0.1 and the episode terminates. If it takes a non-terminal action, it transitions to the next state in the chain and receives a reward of 0.1. Otherwise, if it takes a terminal action, the episode terminates with a reward of 0.
+
+# B.2 TABULAR EXPERIMENTS
+
+During training, we execute each tabular agent in the environment until the episode terminates. Then, we perform a learning step using the experience generated from the previous episode. This process repeats for some number of episodes (in our experiments, 500). After training, we execute each agent in the environment 100 times and compute the average reward achieved across these 100 episodes. For all cases in which search is used, we use a UCT exploration constant of $c = 0 . 1$ .
+
+Q-Learning Tabular Q-learning begins with a table of state-action values initialized to zero. We perform epsilon-greedy exploration with $\epsilon = 0 . 1$ , and add the resulting experience to a replay buffer with maximum size of 1000 transitions. We perform episodic learning, where during each episode the Q-values are fixed and after the episode is complete we update the Q-values by performing a single pass through the experience in the replay buffer in a random order. We use a learning rate of $\beta _ { Q } = 0 . 0 1$ . At test time, the Q-learning agent uses MCTS in the same manner as SAVE.
+
+SAVE Tabular SAVE begins with a table of state-action values initialized to zero. During search, values are looked up in this table and used to initialize $Q _ { 0 }$ . The values are also for bootstrapping. During learning, we perform both Q-learning (as described in the Q-learning agent) as well as an update based on the gradient of the cross-entropy amortization loss (Equation 6). We use $\beta _ { Q } = 0 . 0 1$ and $\beta _ { A } = 1$ .
+
+PUCT Tabular PUCT begins with two tables; one with state values (initialized to zero) and one with action probabilities (initialized to the uniform distribution). During search, action probabilities are looked and used in the PUCT term, while state values are looked up and used for bootstrapping. Search proceeds as described in Section A.3. During learning, $\pi _ { \mathrm { M C T S } }$ is copied back into the action probability table (this is equivalent to an L2 update with a learning rate of 1); we also experimented with doing an update based on the cross entropy loss but found this resulted in worse performance. The value at episode $t$ is given by:
+
+$$
+V _ { t } ( s ) = ( 1 - \alpha ) V _ { t - 1 } ( s ) + \alpha R _ { t - 1 } ( s ) ,
+$$
+
+where $R _ { t - 1 } ( s )$ is the return obtained after visiting state $s$ during episode $t - 1$ . In our experiments we used $\alpha = 0 . 5$ . We also experimented with using Q-learning rather than Monte-Carlo returns, but found that these resulted in similar levels of performance.
+
+UCT The UCT agent is as described in Section 3.1, with $V ( s )$ at unexplored nodes estimated via a Monte-Carlo rollout under a uniform random policy. The only difference from regular UCT is that we did not require all actions to be visited before descending down the search tree; unvisited actions were initialized to a value of zero. For Tightrope, this is the optimal setting of the default Q-values because all possible rewards are greater than or equal to zero. Once an action is found with non-zero reward the best option is to stick with it, so it would not make sense to set the values optimistically. Actions that cause the episode to terminate have a reward of zero, so it would also not make sense to set the values pessimistically as this would lead to over-exploring terminal actions. Setting the values to the average of the parent would either have the effect of setting to zero or setting optimistically (if the parent had positive reward).
+
+To select the final action to execute in the environment, the UCT agent selects a visited action with the maximum estimated value. We could consider alternate approaches here, such as selecting uniformly at random from unexplored actions if none of the visited actions have high enough expected values. We experimented with this approach, using a threshold value of zero (which is the expected value for bad actions in Tightrope), and find that this indeed improves performance $\mathit { p } = 0 . 0 2 )$ , though the effect size is quite small: on the dense setting with $\bar { M } = 9 \bar { 5 } \%$ we achieve a median reward of 0.08 (using this thresholding action selection policy) versus 0.07 (selecting the max of visited actions).
+
+
+Figure C.1: Learning curves on the Covering task. Each plot shows median performance across 10 seeds, with shaded regions showing the min and max seed.
+
+# B.3 FUNCTION APPROXIMATION EXPERIMENTS
+
+We used the same learning setup for the Q-learning, SAVE, and PUCT agents as described in Appendix A. For the network architecture of our agents, we used a shared multilayer perceptron (MLP) torso with two layers of size 64 and ReLU activations. To predict Q-values, we used an MLP head with two layers of size 64 and ReLU activations, with a final layer of size 100 (the number of actions) with a linear activation. To predict a policy in the PUCT agent, we used the same network architecture as the $\mathrm { Q }$ -value head. To predict state values in the PUCT agent, we used a separate MLP head with two layers of size 64 and ReLU activations, and a final layer of size 1 with a linear activation. All network weights were initialized using the default weight initialization scheme in Sonnet (Reynolds et al., 2017). For both the SAVE and PUCT agents we used loss coefficients of $\beta _ { Q } = 0 . 5$ and $\beta _ { A } = 0 . 5$ .
+
+We trained each agent 10 times and report results after 1e6 episodes in a version of Tightrope that has $9 5 \%$ terminal actions Figure 2, right). During training, the SAVE and PUCT agents had access to a search budget of 10 simulations; the Q-learning agent did not use search. We also explored training agents with different numbers of terminal actions and different budgets. Qualitatively, we found the same results as in the tabular setting: the PUCT agent can perform well for larger budgets $( 5 0 + )$ , but struggles with small budgets, underperforming the model-free Q-learning agent. In contrast, SAVE performed well in all our experiments, even for small budgets like 5 or 10.
+
+# C DETAILS ON CONSTRUCTION
+
+# C.1 AGENT DETAILS
+
+SAVE For SAVE, we annealed $\beta _ { Q }$ from 1 to 0.1 and $\beta _ { \mathrm { P I } }$ from 0 to 4.5 over the course of 5e4 episodes. We found this allowed the agent to rely more on Q-learning early on in training to build a good Q-value prior, and then more on MCTS later in training once a good prior had already been established.
+
+
+Figure C.2: Detailed final results on the Covering task. Each plot shows median performance across 10 seeds, with error bars showing the min and max seed.
+
+Q-Learning The Q-Learning agent is as described in Section A.1. In particular, we follow the same setup as the GN-DQN agent described in Bapst et al. (2019). During training, we use pure Q-learning with no search. At test time, we may allow the Q-learning agent to additionally perform MCTS, using the same search procedure as that used by SAVE (i.e., initializing the Q-values using the trained Q-function and initializing the visit counts to one).
+
+SAVE without Amortization Loss The SAVE without an amortization loss is the same as the basic SAVE agent, except that it includes no amortization loss (i.e., $\mathcal { L } ( \boldsymbol { \theta } , \mathcal { D } ) = \beta _ { Q } \mathcal { L } _ { Q } ( \boldsymbol { \theta } , \mathcal { D } ) )$ . This is equivalent to the GN-DQN-MCTS agent described by Bapst et al. (2019).
+
+UCT UCT is as described in Section 3.1, with $V ( s )$ at unexplored nodes estimated via using a pretrained action-value function (trained using the same setup as the Q-learning agent). Additionally, unlike standard UCT we did not require all actions to be visited before descending down the search tree.
+
+SAVE with L2 SAVE with an L2 loss is identical to SAVE except that it uses a different amortizaton loss:
+
+$$
+\mathcal { L } _ { A } ( \theta , \mathcal { D } ) = \frac { 1 } { N } \sum _ { D } \bigl \| Q _ { \mathrm { M C T S } } ( s , \cdot ) - Q _ { \theta } ( s , \cdot ) \bigr \| _ { 2 }
+$$
+
+Similar to the SAVE agent, we anneal $\beta _ { Q }$ from 1 to 0.1 and $\beta _ { A }$ from 0 to 0.045 over the course of 5e4 episodes.
+
+SAVE without Q-Learning SAVE without the Q-learning loss is identical to SAVE except that we do not use Q-learning and we use the L2 amortization loss described in the previous paragraph:
+
+$$
+\mathcal { L } ( \boldsymbol { \theta } , \mathcal { D } ) = \beta _ { A } \mathcal { L } _ { A } ( \boldsymbol { \theta } , \mathcal { D } )
+$$
+
+where we set $\beta _ { A } = 0 . 0 2 5$ . The reason we use the L2 loss rather than the cross-entropy loss is that otherwise the Q-values will not actually be real Q-values, in that they will not have grounding in the actual scale of rewards. We did experiment with using only the cross-entropy loss with no Q-learning, and found slightly worse performance than when using the L2 loss and no Q-learning.
+
+SAVE with PUCT SAVE with PUCT uses the same learning procedure as SAVE but a different search policy. Specifically, we use the PUCT search policy described in Section A.3 and Equation 7. To do this, we set $\pi ( s , a ) = \sigma ( Q _ { \theta } ( s , a ) )$ , where $\sigma$ is the softmax over actions with a temperature of 1. We use the same settings for Dirchlet noise to encourage exploration during search. After search is complete, we select an action using the same epsilon-greedy action procedure used by the SAVE agent rather than selecting based on visit counts. We experimented with selecting based on visit counts instead, but found this resulted in the same level of performance.
+
+# C.2 EXPERIMENTAL SETUP
+
+Observations are given as graphs representing the scene, with objects in the scene corresponding to nodes in the graph and edges between every pair of objects. All agents use the same network architecture (Battaglia et al., 2018) described in Bapst et al. (2019) to process these graphs. Briefly, we use a graph network architecture which takes a graph as input and returns a graph with Q-values on the edges of the graph. Each edge corresponds to a relative object-based action like “pick up block B and put it on block D”. Each edge additionally has multiple actions associated with it which correspond to particular offset locations where the block should be placed, such as “on the top left”.
+
+Bapst et al. (2019) describe four Construction tasks: Silhouette, Connecting, Covering, and Covering Hard. We reported results on three of these tasks in the main text (Connecting, Covering, and Covering Hard). The agents in Bapst et al. (2019) already reached ceiling performance on Silhouette and thus we do not report results for that task here, except to report that SAVE also reaches ceiling performance.
+
+The agents used 10 MCTS simulations during training and were evaluated on 0 to 50 simulations at test time, with the exception of the UCT agent, which always used 1000 simulations at test time, and the Q-learning agent, which did not peform search during learning. We trained 10 seeds per agent and report results after 1e6 episodes. Figure C.1 show details of learning progress for each of the agents compared in the ablation experiments on the Covering task (Section 4.2), and Figure C.2 shows detailed final performances evaluated at different test budgets. We evaluated all agents on the hardest level of difficulty of the particular task they were trained on for either 10000 episodes (Figure 3a-c) or 1000 episodes (Figure 3d and Figure C.2). In general, while we find that search at test time can provide small boosts in performance, the main gains are achieved by incorporating search during training.
+
+# C.3 DISCUSSION OF ABLATION RESULTS
+
+Here we expand on the results presented in the main text and in Figure 3d and Figure C.2.
+
+Cross-entropy vs. L2 loss While the L2 loss (Figure C.2, orange) can result in equivalent performance as the cross-entropy loss (Figure C.2, green), this is at the cost of higher variance across seeds and lower performance on average. This is likely because the L2 loss encourages the Q-function to exactly match the Q-values estimated by search. However, with a search budget of 10, those Qvalues will be very noisy. In contrast, the cross-entropy loss only encourages the Q-function to match the overall distribution shape of the Q-values estimated by search. This is a less strong constraint that allows the information acquired during search to be exploited while not relying on it too strongly. Indeed, we can observe that the agent with L2 amortization loss actually performs worse than the agent that has no amortization loss at all (Figure C.2, purple) when using a search budget of 10, suggesting that trying to match the Q-values during search too closely can harm performance.
+
+Additionally, we can consider an interesting interaction between Q-learning and the amortization loss. Due to the search locally avoiding poor actions, Q-learning will rarely actually operate on low-valued actions, meaning most of its computation is spent refining the estimates for high-valued actions. The softmax cross entropy loss ensures that low-valued actions have lower values than high-valued actions, but does not force these values to be exact. Thus, in this regime we should have good estimates of value for high-valued actions and worse estimates of value for low-valued actions. In contrast, an L2 loss would require the values to be exact for both low and high valued actions. By using cross entropy instead, we can allow the neural network to spend more of its capacity representing the high-valued actions and less capacity representing the low-valued actions, which we care less about in the first place anyway.
+
+With vs. without Q-learning Without Q-learning (Figure C.2, teal), the SAVE agent’s performance suffers dramatically. As discussed in the previous section, the Q-values estimated during search are very noisy, meaning it is not necessarily a good idea to try to match them exactly. Additionally, $Q _ { \mathrm { M C T S } }$ is on-policy experience and can become stale if $Q _ { \theta }$ changes too much between when $Q _ { \mathrm { M C T S } }$ was computed and when it is used for learning. Thus, removing the Q-learning loss makes the learning algorithm much more on-policy and therefore susceptible to the issues that come with on-policy training. Indeed, without the Q-learning loss, we can only rely on the Q-values estimated during search, resulting in much worse performance than when Q-learning is used.
+
+
+Figure C.3: Performance of different exploration strategies on the Covering task.
+
+
+Figure C.4: Performance of SAVE and Q-learning on Covering, controlling for the same number of environment interactions (including those seen during search).
+
+UCT vs. PUCT Finally, we compared to a variant which utilizes prior knowledge by transforming the Q-values into a policy via a softmax and then using this policy as a prior with PUCT, rather than using it to initialize the Q-values (Figure C.2, brown). With large amounts of search, the initial setting of the Q-values should not matter much, but in the case of small search budgets (as seen here), the estimated Q-values do not change much from their initial values. Thus, if the initial values are zero, then the final values will also be close to zero, which later results in the Q-function being regressed towards a nearly uniform distribution of value. By initializing the Q-values with the Qfunction, the values that are regressed towards may be similar to the original Q-function but will not be uniform. Thus, we can more effectively reuse knowledge across multiple searches by initializing the Q-values with UCT rather than incorporating prior knowledge via PUCT.
+
+# C.4 ADDITIONAL RESULTS
+
+We performed several other experiments to tease apart the questions regarding exploration strategy and data efficiency.
+
+Exploration strategy When selecting the final action to perform in the environment, SAVE uses an epsilon-greedy exploration strategy. However, many other exploration strategies might be considered, such as UCB, categorical sampling from the softmax of estimated Q-values, or categorical sampling from the normalized visit counts. We evaluated how well each of these exploration strategies work, with the results shown in Figure C.3. We find that using epsilon-greedy works the best out of these exploration strategies by a substantial margin. We speculate that this may be because it is important for the Q-function to be well approximated across all actions, so that it is useful during MCTS backups. However, UCB and categorical methods will not uniformly sample the action space, meaning that some actions are very unlikely to be ever learned from. The amortization loss will not help either, as these actions will not be explored during search either. The error in the Q-values for unexplored actions will grow over time (due to catastrophic forgetting), leading to a poorly approximated Q-function that is unreliable. In contrast, epsilon-greedy consistently spends a little bit of time exploring these actions, preventing their values from becoming too inaccurate. We expect this would be less of a problem if we were to use a separate state-value function for bootstrapping (as is done by AlphaZero).
+
+Data efficiency With a search budget of 10, SAVE effectively sees 10 times as many transitions as a model-free agent trained on the same number of environment interactions. To more carefully compare the data efficiency of SAVE, we compared its performance to that of the Q-learning agent on the Covering task, controlling for the same number of environment interactions (including those seen during search). The results are shown in Figure C.4, illustrating that SAVE converges to higher rewards given the same amount of data. We find similar results in the Marble Run environment, shown in Figure D.2.
+
+# D DETAILS ON MARBLE RUN
+
+# D.1 SCENE GENERATION
+
+Scenes contain the following types of objects (similar to Bapst et al. (2019)):
+
+• Floor (in black) that supports the blocks placed by the agent.
+• Available blocks (row of blue blocks at the bottom) that the agent picks and place in the scene (with replacement).
+Blocks (blue blocks above the floor) that the agent has already placed. They may take a lighter blue color to indicate that they are sticky. A sticky block gets glued to anything it touches.
+• Goal (blue dot) that the agent has to reach with the marble.
+• Marble (green circle) that the agent has to route to the goal.
+• Obstacles (red blocks, including two vertical walls), that the agent has to avoid, by not touching them neither with the blocks or the marble.
+
+All the initial positions for obstacles in the scene are sampled from a tessellation (similar to the Silhouette task in Bapst et al. (2019)) made of rows with random sequences of blocks with sizes of 1 discretization unit in height and 1 or 2 discretization units in width (a discretization unit corresponds to the side of the first available block). The sampling process goes as follows:
+
+1. Set the vertical position of the goal to the specified discrete height (according to level) corresponding to the center of one of the tessellation rows, and the vertical position of the marble 2 rows above that.
+2. Uniformly sample a horizontal distance between the marble and the goal from a predefined range, and uniformly sample the absolute horizontal positions respecting that absolute distance.
+3. Sample a number of obstacles (according to level) from the tessellation spanning up to the vertical position of the marble.
+
+Obstacles are sampled from the tessellation sequentially. Before each obstacle is sampled, all objects in the tessellation that are too close ( $\pm 2$ layers vertically and with less than 2 discretization units of clearance sideways) to the goal, the target, or previously placed obstacles, are removed from the tessellation in order to prevent unsolvable scenes. Then probabilities are assigned to all of the remaining objects in the tessellation according to one of the following criteria (the criteria itself is also picked randomly with different weights) designed to avoid generating trivial scenes:
+
+• (Weigh $^ { - 4 }$ ) Pick uniformly a tessellation object lying exactly on the floor and between the marble and the goal horizontally, since those objects prevent the marble from rolling freely on the floor (only applicable if the tessellation still has objects of this kind available).
+
+• (Weight=1) Pick a tessellation object that is close (horizontally) to the marble. Probabilities proportional to $\frac { 1 } { ( d / \tau ) ^ { 2 } + 0 . 1 }$ (where $d$ is the horizontal distance between each object and the marble scaled by the width of the scene and $\tau$ is a temperature set to 0.1) are assigned to all objects left in the tessellation, and one of them is picked. (Weight=1) Pick a tessellation object that is close (horizontally) to the goal. Identical to the previous one, but using the distance to the goal. (Weight=1) Pick a tessellation object that is close (horizontally) to the middle point between the ball and the goal. Identical to the previous one, but using the distance to the middle point, and a temperature of 0.2.
+• (Weigh $^ { = 1 }$ ) Pick any object remaining in the tessellation with uniform probability (to increase diversity).
+
+# D.2 CURRICULUM DIFFICULTY
+
+We used a curriculum to sample scenes of increasing difficulty (Fig. D.1) according to:
+
+| Algorithm A.1 Pseudocode for the SAVE algorithm. |
| 1: procedure SAVE(θ) |
| 2: | while true do |
| 3: | Begin episode at s |
| 4: | while acting do |
| 5: | Estimate QmCTs(s,:) ← MCTS(s, Qθ) |
| 6: | Select a using epsilon-greedy from QMCTs(s,:) |
| 7: 8: | Execute a in environment and receive s',r |
| 9: | Add (s,a,r,s',QmCTs(s,·)) to replay buffer |
| s↑s` |
| 10: | while learning do |
| 11: | Sample minibatch of experience from the replay buffer |
| 12: | Update θ to minimize Equation 6 |
| 13: |
| 14: | procedure MCTS(so, Qθ) |
| 15: | Qo(s,a)←Qe(s,a) forall s,a |
| 16: | No(s,a) ←1for all s,a |
| 17: | k←0 |
| 18: 19: | while search budget remains (k < K) do |
| 20: | Traverse the search tree with πk (Equation 1) |
| 21: | Expand new state sT and add it to the search tree |
| 22: | Evaluate maxa Qe(sT,a) and backup returns (Equation 3) |
| 23: | Set Nk+1(s,a) ← Nk(s,a) and then increment counts of visited states and actions |
| Compute estimates for Qk+1(s,a) (Equation 4) |
| 24: | k←k+1 |
| 25: | Return {Qk(so,ai)}i |
",
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+ {
+ "type": "text",
+ "text": "The SAVE agent is implemented as described in Section 3 and Algorithm A.1 provides additional pseudocode explaining the algorithm. In Algorithm A.1, we provide an example of using SAVE in an episode setting where learning happens after every episode; however, SAVE can be used in any Q-learning setup including in distributed setups where separate processes are concurrently acting and learning. In particular, in our experiments we use the distributed setup described in Section A.1. Note that when performing epsilon-greedy exploration (Line 6 of Algorithm A.1), we either choose an action uniformly at random with probability $\\epsilon$ , and otherwise choose the action with the highest value of $Q _ { \\mathrm { M C T S } }$ out of the actions which were explored during search (i.e., we do not consider actions that were not explored, even if they have a higher $Q _ { \\mathrm { M C T S } } )$ . In all experiments (except tabular Tightrope), we use a UTC exploration constant of $c = 2$ , though we have found SAVE’s performance to be relatively robust to this parameter setting. ",
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+ "type": "text",
+ "text": "A.3 PUCT ",
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+ "text": "The PUCT search policy is based on that described by Silver et al. (2017a) and Silver et al. (2018). Specifically, we choose actions during search according to Equation 1, with: ",
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+ "img_path": "images/4cae67b2ef3fe96812d333da76451631826e4eb1c606c2c6efec07d7e888de2f.jpg",
+ "text": "$$\n\\begin{array} { c } { { Q _ { k } = \\displaystyle \\frac { \\sum _ { i = 1 } ^ { N _ { k } ( s , a ) } R _ { i } ( s , a ) } { N _ { k } ( s , a ) } } } \\\\ { { { } } } \\\\ { { U _ { k } ( s , a ) = c \\cdot \\pi ( s , a ) \\displaystyle \\frac { \\sqrt { \\sum _ { a } N _ { k } ( s , a ) } } { N _ { k } ( s , a ) + 1 } } } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "where $c$ is an exploration constant, $\\pi ( s , a )$ is the prior policy, and $N _ { k } ( s , a )$ is the total number of times action $a$ had been taken from state $s$ at iteration $k$ of the search. Like Silver et al. (2017a; 2018), we add Dirichlet noise to the prior policy: ",
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+ {
+ "type": "equation",
+ "img_path": "images/b97e9e54b23a61cf344d6299d878b4d0cd8011bcd6ce943e38632e5e75515a93.jpg",
+ "text": "$$\n\\pi ( s , a ) = ( 1 - \\epsilon ) \\cdot \\pi _ { \\theta } ( s , a ) + \\epsilon \\eta ,\n$$",
+ "text_format": "latex",
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+ "text": "where $\\eta \\sim \\mathrm { D i r } ( 1 / n _ { \\mathrm { a c t i o n s } } )$ . In our experiments we set $\\epsilon = 0 . 2 5$ and $c = 2$ . During training, after search is complete, we sample an action to execute in the environment from $\\pi _ { \\mathrm { M C T S } } ( s _ { 0 } , a ) =$ $\\begin{array} { r } { N _ { K } ( s _ { 0 } , a ) / \\sum _ { a } { N _ { K } ^ { - } ( s _ { 0 } , a ) } } \\end{array}$ . At test time, we select the action which has the maximum visit count (with random tie-breaking). ",
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+ "type": "text",
+ "text": "To train the PUCT agent, we used separate policy $\\pi _ { \\theta } ( s , a )$ and value $V _ { \\theta } ( s )$ heads which were trained using a combined loss (Equation 6), with: ",
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+ {
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+ "img_path": "images/8f03f07d46b0068bac9cb20506410780ed7b6669ad7380ff549d2a6b65537cd1.jpg",
+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } _ { Q } = \\frac { 1 } { N } \\sum _ { \\mathcal { D } } \\big \\| V _ { \\boldsymbol { \\theta } } ( s ) - R \\big \\| _ { 2 } } \\\\ { \\displaystyle \\mathcal { L } _ { A } = - \\frac { 1 } { N } \\sum _ { \\mathcal { D } } \\pi _ { \\mathrm { M C T S } } ( s , \\cdot ) ^ { \\top } \\log \\pi _ { \\boldsymbol { \\theta } } ( s , \\cdot ) } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "where $R$ is the Monte-Carlo return observed from state $s$ . We used fixed values of $\\beta _ { Q } = 0 . 5$ and $\\beta _ { A } = 0 . 5$ in all our experiments with PUCT. We used the same replay and training setup as used in the Q-learning and SAVE agents, with two exceptions. First, we additionally include episodic Monte-Carlo returns $R$ and policies $\\pi _ { \\mathrm { M C T S } }$ in the replay buffer so they can be used during learning. Second, we did not use $\\epsilon$ -greedy exploration (because the Dirichlet noise in the PUCT term already enables sufficient exploration). ",
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+ "type": "text",
+ "text": "We tried several different hyperparameter settings and variants of the PUCT agent to attempt to improve the results. For example, we tried using a 1-step TD error for learning the values, which should have lower variance and thus result in more stable learning of values. We also tried reducing the replay ratio to 1 and the replay size to 400 in order to make the experience for training more on-policy. However, we did not find that these changes improved the results. We also tried different settings of $\\epsilon$ for the Dirichlet noise, but found that lower values resulted in too little exploration, while higher values resulted in too much exploration. ",
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+ },
+ {
+ "type": "text",
+ "text": "B DETAILS ON TIGHTROPE",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "B.1 ENVIRONMENT ",
+ "text_level": 1,
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+ },
+ {
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+ "text": "The Tightrope environment has 11 states which are connected together in a chain. Each state has 100 actions, $M \\%$ of which will cause the episode to terminate when executed and the rest of which will cause the environment to transition to the next state. Each state is represented using a vector of 50 random values drawn from a standard normal distribution, which are the same across episodes. The indices of terminal actions are selected randomly and are different for each state but are consistent across episodes. Agents always begin in the first state of the chain. ",
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+ },
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+ "type": "text",
+ "text": "In the sparse reward setting, we randomly select one of the states in the chain to be the “final” state (excluding the first state), to enable the agent to sometimes train on easy problems and sometimes train on hard problems. If the agent reaches this final state, it receives a reward of 1 and the episode terminates. If it takes a non-terminal action, it transitions to the next state in the chain and receives a reward of 0. Otherwise, if it takes a terminal action, the episode terminates and the agent receives a reward of 0. ",
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+ "type": "text",
+ "text": "In the dense reward setting, the “final” state is always chosen to be the last state in the chain. If the agent reaches the final state in the chain, it receives a reward of 0.1 and the episode terminates. If it takes a non-terminal action, it transitions to the next state in the chain and receives a reward of 0.1. Otherwise, if it takes a terminal action, the episode terminates with a reward of 0. ",
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+ {
+ "type": "text",
+ "text": "B.2 TABULAR EXPERIMENTS ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "During training, we execute each tabular agent in the environment until the episode terminates. Then, we perform a learning step using the experience generated from the previous episode. This process repeats for some number of episodes (in our experiments, 500). After training, we execute each agent in the environment 100 times and compute the average reward achieved across these 100 episodes. For all cases in which search is used, we use a UCT exploration constant of $c = 0 . 1$ . ",
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+ "text": "Q-Learning Tabular Q-learning begins with a table of state-action values initialized to zero. We perform epsilon-greedy exploration with $\\epsilon = 0 . 1$ , and add the resulting experience to a replay buffer with maximum size of 1000 transitions. We perform episodic learning, where during each episode the Q-values are fixed and after the episode is complete we update the Q-values by performing a single pass through the experience in the replay buffer in a random order. We use a learning rate of $\\beta _ { Q } = 0 . 0 1$ . At test time, the Q-learning agent uses MCTS in the same manner as SAVE. ",
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+ "type": "text",
+ "text": "SAVE Tabular SAVE begins with a table of state-action values initialized to zero. During search, values are looked up in this table and used to initialize $Q _ { 0 }$ . The values are also for bootstrapping. During learning, we perform both Q-learning (as described in the Q-learning agent) as well as an update based on the gradient of the cross-entropy amortization loss (Equation 6). We use $\\beta _ { Q } = 0 . 0 1$ and $\\beta _ { A } = 1$ . ",
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+ {
+ "type": "text",
+ "text": "PUCT Tabular PUCT begins with two tables; one with state values (initialized to zero) and one with action probabilities (initialized to the uniform distribution). During search, action probabilities are looked and used in the PUCT term, while state values are looked up and used for bootstrapping. Search proceeds as described in Section A.3. During learning, $\\pi _ { \\mathrm { M C T S } }$ is copied back into the action probability table (this is equivalent to an L2 update with a learning rate of 1); we also experimented with doing an update based on the cross entropy loss but found this resulted in worse performance. The value at episode $t$ is given by: ",
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+ "type": "equation",
+ "img_path": "images/cd230dd65124296c8633ed504fcd83d980349352b61f832d054993ed33249cd7.jpg",
+ "text": "$$\nV _ { t } ( s ) = ( 1 - \\alpha ) V _ { t - 1 } ( s ) + \\alpha R _ { t - 1 } ( s ) ,\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "where $R _ { t - 1 } ( s )$ is the return obtained after visiting state $s$ during episode $t - 1$ . In our experiments we used $\\alpha = 0 . 5$ . We also experimented with using Q-learning rather than Monte-Carlo returns, but found that these resulted in similar levels of performance. ",
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+ "text": "UCT The UCT agent is as described in Section 3.1, with $V ( s )$ at unexplored nodes estimated via a Monte-Carlo rollout under a uniform random policy. The only difference from regular UCT is that we did not require all actions to be visited before descending down the search tree; unvisited actions were initialized to a value of zero. For Tightrope, this is the optimal setting of the default Q-values because all possible rewards are greater than or equal to zero. Once an action is found with non-zero reward the best option is to stick with it, so it would not make sense to set the values optimistically. Actions that cause the episode to terminate have a reward of zero, so it would also not make sense to set the values pessimistically as this would lead to over-exploring terminal actions. Setting the values to the average of the parent would either have the effect of setting to zero or setting optimistically (if the parent had positive reward). ",
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+ "type": "text",
+ "text": "To select the final action to execute in the environment, the UCT agent selects a visited action with the maximum estimated value. We could consider alternate approaches here, such as selecting uniformly at random from unexplored actions if none of the visited actions have high enough expected values. We experimented with this approach, using a threshold value of zero (which is the expected value for bad actions in Tightrope), and find that this indeed improves performance $\\mathit { p } = 0 . 0 2 )$ , though the effect size is quite small: on the dense setting with $\\bar { M } = 9 \\bar { 5 } \\%$ we achieve a median reward of 0.08 (using this thresholding action selection policy) versus 0.07 (selecting the max of visited actions). ",
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+ "type": "image",
+ "img_path": "images/69e2b140907abcc2661c55e2c7a60c7d8314c127da7a48db91fa330dfabf6c29.jpg",
+ "image_caption": [
+ "Figure C.1: Learning curves on the Covering task. Each plot shows median performance across 10 seeds, with shaded regions showing the min and max seed. "
+ ],
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+ "type": "text",
+ "text": "B.3 FUNCTION APPROXIMATION EXPERIMENTS ",
+ "text_level": 1,
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+ "text": "We used the same learning setup for the Q-learning, SAVE, and PUCT agents as described in Appendix A. For the network architecture of our agents, we used a shared multilayer perceptron (MLP) torso with two layers of size 64 and ReLU activations. To predict Q-values, we used an MLP head with two layers of size 64 and ReLU activations, with a final layer of size 100 (the number of actions) with a linear activation. To predict a policy in the PUCT agent, we used the same network architecture as the $\\mathrm { Q }$ -value head. To predict state values in the PUCT agent, we used a separate MLP head with two layers of size 64 and ReLU activations, and a final layer of size 1 with a linear activation. All network weights were initialized using the default weight initialization scheme in Sonnet (Reynolds et al., 2017). For both the SAVE and PUCT agents we used loss coefficients of $\\beta _ { Q } = 0 . 5$ and $\\beta _ { A } = 0 . 5$ . ",
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+ "type": "text",
+ "text": "We trained each agent 10 times and report results after 1e6 episodes in a version of Tightrope that has $9 5 \\%$ terminal actions Figure 2, right). During training, the SAVE and PUCT agents had access to a search budget of 10 simulations; the Q-learning agent did not use search. We also explored training agents with different numbers of terminal actions and different budgets. Qualitatively, we found the same results as in the tabular setting: the PUCT agent can perform well for larger budgets $( 5 0 + )$ , but struggles with small budgets, underperforming the model-free Q-learning agent. In contrast, SAVE performed well in all our experiments, even for small budgets like 5 or 10. ",
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+ "type": "text",
+ "text": "C DETAILS ON CONSTRUCTION ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "C.1 AGENT DETAILS ",
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+ "type": "text",
+ "text": "SAVE For SAVE, we annealed $\\beta _ { Q }$ from 1 to 0.1 and $\\beta _ { \\mathrm { P I } }$ from 0 to 4.5 over the course of 5e4 episodes. We found this allowed the agent to rely more on Q-learning early on in training to build a good Q-value prior, and then more on MCTS later in training once a good prior had already been established. ",
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+ "img_path": "images/8360ebb11c4809ad4104a610431719b8e0e3c6323295ebdd44cd59dea77e01b7.jpg",
+ "image_caption": [
+ "Figure C.2: Detailed final results on the Covering task. Each plot shows median performance across 10 seeds, with error bars showing the min and max seed. "
+ ],
+ "image_footnote": [],
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+ {
+ "type": "text",
+ "text": "Q-Learning The Q-Learning agent is as described in Section A.1. In particular, we follow the same setup as the GN-DQN agent described in Bapst et al. (2019). During training, we use pure Q-learning with no search. At test time, we may allow the Q-learning agent to additionally perform MCTS, using the same search procedure as that used by SAVE (i.e., initializing the Q-values using the trained Q-function and initializing the visit counts to one). ",
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+ "text": "SAVE without Amortization Loss The SAVE without an amortization loss is the same as the basic SAVE agent, except that it includes no amortization loss (i.e., $\\mathcal { L } ( \\boldsymbol { \\theta } , \\mathcal { D } ) = \\beta _ { Q } \\mathcal { L } _ { Q } ( \\boldsymbol { \\theta } , \\mathcal { D } ) )$ . This is equivalent to the GN-DQN-MCTS agent described by Bapst et al. (2019). ",
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+ "text": "UCT UCT is as described in Section 3.1, with $V ( s )$ at unexplored nodes estimated via using a pretrained action-value function (trained using the same setup as the Q-learning agent). Additionally, unlike standard UCT we did not require all actions to be visited before descending down the search tree. ",
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+ "text": "SAVE with L2 SAVE with an L2 loss is identical to SAVE except that it uses a different amortizaton loss: ",
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+ "text": "$$\n\\mathcal { L } _ { A } ( \\theta , \\mathcal { D } ) = \\frac { 1 } { N } \\sum _ { D } \\bigl \\| Q _ { \\mathrm { M C T S } } ( s , \\cdot ) - Q _ { \\theta } ( s , \\cdot ) \\bigr \\| _ { 2 }\n$$",
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+ "text": "Similar to the SAVE agent, we anneal $\\beta _ { Q }$ from 1 to 0.1 and $\\beta _ { A }$ from 0 to 0.045 over the course of 5e4 episodes. ",
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+ "text": "SAVE without Q-Learning SAVE without the Q-learning loss is identical to SAVE except that we do not use Q-learning and we use the L2 amortization loss described in the previous paragraph: ",
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+ "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } , \\mathcal { D } ) = \\beta _ { A } \\mathcal { L } _ { A } ( \\boldsymbol { \\theta } , \\mathcal { D } )\n$$",
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+ "text": "where we set $\\beta _ { A } = 0 . 0 2 5$ . The reason we use the L2 loss rather than the cross-entropy loss is that otherwise the Q-values will not actually be real Q-values, in that they will not have grounding in the actual scale of rewards. We did experiment with using only the cross-entropy loss with no Q-learning, and found slightly worse performance than when using the L2 loss and no Q-learning. ",
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+ "text": "SAVE with PUCT SAVE with PUCT uses the same learning procedure as SAVE but a different search policy. Specifically, we use the PUCT search policy described in Section A.3 and Equation 7. To do this, we set $\\pi ( s , a ) = \\sigma ( Q _ { \\theta } ( s , a ) )$ , where $\\sigma$ is the softmax over actions with a temperature of 1. We use the same settings for Dirchlet noise to encourage exploration during search. After search is complete, we select an action using the same epsilon-greedy action procedure used by the SAVE agent rather than selecting based on visit counts. We experimented with selecting based on visit counts instead, but found this resulted in the same level of performance. ",
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+ "text": "C.2 EXPERIMENTAL SETUP ",
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+ "text": "Observations are given as graphs representing the scene, with objects in the scene corresponding to nodes in the graph and edges between every pair of objects. All agents use the same network architecture (Battaglia et al., 2018) described in Bapst et al. (2019) to process these graphs. Briefly, we use a graph network architecture which takes a graph as input and returns a graph with Q-values on the edges of the graph. Each edge corresponds to a relative object-based action like “pick up block B and put it on block D”. Each edge additionally has multiple actions associated with it which correspond to particular offset locations where the block should be placed, such as “on the top left”. ",
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+ "text": "Bapst et al. (2019) describe four Construction tasks: Silhouette, Connecting, Covering, and Covering Hard. We reported results on three of these tasks in the main text (Connecting, Covering, and Covering Hard). The agents in Bapst et al. (2019) already reached ceiling performance on Silhouette and thus we do not report results for that task here, except to report that SAVE also reaches ceiling performance. ",
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+ "text": "The agents used 10 MCTS simulations during training and were evaluated on 0 to 50 simulations at test time, with the exception of the UCT agent, which always used 1000 simulations at test time, and the Q-learning agent, which did not peform search during learning. We trained 10 seeds per agent and report results after 1e6 episodes. Figure C.1 show details of learning progress for each of the agents compared in the ablation experiments on the Covering task (Section 4.2), and Figure C.2 shows detailed final performances evaluated at different test budgets. We evaluated all agents on the hardest level of difficulty of the particular task they were trained on for either 10000 episodes (Figure 3a-c) or 1000 episodes (Figure 3d and Figure C.2). In general, while we find that search at test time can provide small boosts in performance, the main gains are achieved by incorporating search during training. ",
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+ "text": "C.3 DISCUSSION OF ABLATION RESULTS ",
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+ "text": "Here we expand on the results presented in the main text and in Figure 3d and Figure C.2. ",
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+ "text": "Cross-entropy vs. L2 loss While the L2 loss (Figure C.2, orange) can result in equivalent performance as the cross-entropy loss (Figure C.2, green), this is at the cost of higher variance across seeds and lower performance on average. This is likely because the L2 loss encourages the Q-function to exactly match the Q-values estimated by search. However, with a search budget of 10, those Qvalues will be very noisy. In contrast, the cross-entropy loss only encourages the Q-function to match the overall distribution shape of the Q-values estimated by search. This is a less strong constraint that allows the information acquired during search to be exploited while not relying on it too strongly. Indeed, we can observe that the agent with L2 amortization loss actually performs worse than the agent that has no amortization loss at all (Figure C.2, purple) when using a search budget of 10, suggesting that trying to match the Q-values during search too closely can harm performance. ",
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+ "text": "Additionally, we can consider an interesting interaction between Q-learning and the amortization loss. Due to the search locally avoiding poor actions, Q-learning will rarely actually operate on low-valued actions, meaning most of its computation is spent refining the estimates for high-valued actions. The softmax cross entropy loss ensures that low-valued actions have lower values than high-valued actions, but does not force these values to be exact. Thus, in this regime we should have good estimates of value for high-valued actions and worse estimates of value for low-valued actions. In contrast, an L2 loss would require the values to be exact for both low and high valued actions. By using cross entropy instead, we can allow the neural network to spend more of its capacity representing the high-valued actions and less capacity representing the low-valued actions, which we care less about in the first place anyway. ",
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+ "text": "With vs. without Q-learning Without Q-learning (Figure C.2, teal), the SAVE agent’s performance suffers dramatically. As discussed in the previous section, the Q-values estimated during search are very noisy, meaning it is not necessarily a good idea to try to match them exactly. Additionally, $Q _ { \\mathrm { M C T S } }$ is on-policy experience and can become stale if $Q _ { \\theta }$ changes too much between when $Q _ { \\mathrm { M C T S } }$ was computed and when it is used for learning. Thus, removing the Q-learning loss makes the learning algorithm much more on-policy and therefore susceptible to the issues that come with on-policy training. Indeed, without the Q-learning loss, we can only rely on the Q-values estimated during search, resulting in much worse performance than when Q-learning is used. ",
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+ "Figure C.3: Performance of different exploration strategies on the Covering task. "
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+ "Figure C.4: Performance of SAVE and Q-learning on Covering, controlling for the same number of environment interactions (including those seen during search). "
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+ "text": "UCT vs. PUCT Finally, we compared to a variant which utilizes prior knowledge by transforming the Q-values into a policy via a softmax and then using this policy as a prior with PUCT, rather than using it to initialize the Q-values (Figure C.2, brown). With large amounts of search, the initial setting of the Q-values should not matter much, but in the case of small search budgets (as seen here), the estimated Q-values do not change much from their initial values. Thus, if the initial values are zero, then the final values will also be close to zero, which later results in the Q-function being regressed towards a nearly uniform distribution of value. By initializing the Q-values with the Qfunction, the values that are regressed towards may be similar to the original Q-function but will not be uniform. Thus, we can more effectively reuse knowledge across multiple searches by initializing the Q-values with UCT rather than incorporating prior knowledge via PUCT. ",
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+ "text": "C.4 ADDITIONAL RESULTS ",
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+ "text": "We performed several other experiments to tease apart the questions regarding exploration strategy and data efficiency. ",
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+ "text": "Exploration strategy When selecting the final action to perform in the environment, SAVE uses an epsilon-greedy exploration strategy. However, many other exploration strategies might be considered, such as UCB, categorical sampling from the softmax of estimated Q-values, or categorical sampling from the normalized visit counts. We evaluated how well each of these exploration strategies work, with the results shown in Figure C.3. We find that using epsilon-greedy works the best out of these exploration strategies by a substantial margin. We speculate that this may be because it is important for the Q-function to be well approximated across all actions, so that it is useful during MCTS backups. However, UCB and categorical methods will not uniformly sample the action space, meaning that some actions are very unlikely to be ever learned from. The amortization loss will not help either, as these actions will not be explored during search either. The error in the Q-values for unexplored actions will grow over time (due to catastrophic forgetting), leading to a poorly approximated Q-function that is unreliable. In contrast, epsilon-greedy consistently spends a little bit of time exploring these actions, preventing their values from becoming too inaccurate. We expect this would be less of a problem if we were to use a separate state-value function for bootstrapping (as is done by AlphaZero). ",
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+ "text": "Data efficiency With a search budget of 10, SAVE effectively sees 10 times as many transitions as a model-free agent trained on the same number of environment interactions. To more carefully compare the data efficiency of SAVE, we compared its performance to that of the Q-learning agent on the Covering task, controlling for the same number of environment interactions (including those seen during search). The results are shown in Figure C.4, illustrating that SAVE converges to higher rewards given the same amount of data. We find similar results in the Marble Run environment, shown in Figure D.2. ",
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+ "text": "D DETAILS ON MARBLE RUN ",
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+ "text": "Scenes contain the following types of objects (similar to Bapst et al. (2019)): ",
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+ "text": "• Floor (in black) that supports the blocks placed by the agent. \n• Available blocks (row of blue blocks at the bottom) that the agent picks and place in the scene (with replacement). \nBlocks (blue blocks above the floor) that the agent has already placed. They may take a lighter blue color to indicate that they are sticky. A sticky block gets glued to anything it touches. \n• Goal (blue dot) that the agent has to reach with the marble. \n• Marble (green circle) that the agent has to route to the goal. \n• Obstacles (red blocks, including two vertical walls), that the agent has to avoid, by not touching them neither with the blocks or the marble. ",
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+ "text": "All the initial positions for obstacles in the scene are sampled from a tessellation (similar to the Silhouette task in Bapst et al. (2019)) made of rows with random sequences of blocks with sizes of 1 discretization unit in height and 1 or 2 discretization units in width (a discretization unit corresponds to the side of the first available block). The sampling process goes as follows: ",
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+ "text": "1. Set the vertical position of the goal to the specified discrete height (according to level) corresponding to the center of one of the tessellation rows, and the vertical position of the marble 2 rows above that. \n2. Uniformly sample a horizontal distance between the marble and the goal from a predefined range, and uniformly sample the absolute horizontal positions respecting that absolute distance. \n3. Sample a number of obstacles (according to level) from the tessellation spanning up to the vertical position of the marble. ",
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+ "text": "Obstacles are sampled from the tessellation sequentially. Before each obstacle is sampled, all objects in the tessellation that are too close ( $\\pm 2$ layers vertically and with less than 2 discretization units of clearance sideways) to the goal, the target, or previously placed obstacles, are removed from the tessellation in order to prevent unsolvable scenes. Then probabilities are assigned to all of the remaining objects in the tessellation according to one of the following criteria (the criteria itself is also picked randomly with different weights) designed to avoid generating trivial scenes: ",
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+ "text": "• (Weigh $^ { - 4 }$ ) Pick uniformly a tessellation object lying exactly on the floor and between the marble and the goal horizontally, since those objects prevent the marble from rolling freely on the floor (only applicable if the tessellation still has objects of this kind available). ",
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+ "text": "• (Weight=1) Pick a tessellation object that is close (horizontally) to the marble. Probabilities proportional to $\\frac { 1 } { ( d / \\tau ) ^ { 2 } + 0 . 1 }$ (where $d$ is the horizontal distance between each object and the marble scaled by the width of the scene and $\\tau$ is a temperature set to 0.1) are assigned to all objects left in the tessellation, and one of them is picked. (Weight=1) Pick a tessellation object that is close (horizontally) to the goal. Identical to the previous one, but using the distance to the goal. (Weight=1) Pick a tessellation object that is close (horizontally) to the middle point between the ball and the goal. Identical to the previous one, but using the distance to the middle point, and a temperature of 0.2. \n• (Weigh $^ { = 1 }$ ) Pick any object remaining in the tessellation with uniform probability (to increase diversity). ",
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+ "table_body": "| Algorithm A.1 Pseudocode for the SAVE algorithm. |
| 1: procedure SAVE(θ) |
| 2: | while true do |
| 3: | Begin episode at s |
| 4: | while acting do |
| 5: | Estimate QmCTs(s,:) ← MCTS(s, Qθ) |
| 6: | Select a using epsilon-greedy from QMCTs(s,:) |
| 7: 8: | Execute a in environment and receive s',r |
| 9: | Add (s,a,r,s',QmCTs(s,·)) to replay buffer |
| s↑s` |
| 10: | while learning do |
| 11: | Sample minibatch of experience from the replay buffer |
| 12: | Update θ to minimize Equation 6 |
| 13: |
| 14: | procedure MCTS(so, Qθ) |
| 15: | Qo(s,a)←Qe(s,a) forall s,a |
| 16: | No(s,a) ←1for all s,a |
| 17: | k←0 |
| 18: 19: | while search budget remains (k < K) do |
| 20: | Traverse the search tree with πk (Equation 1) |
| 21: | Expand new state sT and add it to the search tree |
| 22: | Evaluate maxa Qe(sT,a) and backup returns (Equation 3) |
| 23: | Set Nk+1(s,a) ← Nk(s,a) and then increment counts of visited states and actions |
| Compute estimates for Qk+1(s,a) (Equation 4) |
| 24: | k←k+1 |
| 25: | Return {Qk(so,ai)}i |
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| 1: procedure SAVE(θ) |
| 2: | while true do |
| 3: | Begin episode at s |
| 4: | while acting do |
| 5: | Estimate QmCTs(s,:) ← MCTS(s, Qθ) |
| 6: | Select a using epsilon-greedy from QMCTs(s,:) |
| 7: 8: | Execute a in environment and receive s',r |
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| s↑s` |
| 10: | while learning do |
| 11: | Sample minibatch of experience from the replay buffer |
| 12: | Update θ to minimize Equation 6 |
| 13: |
| 14: | procedure MCTS(so, Qθ) |
| 15: | Qo(s,a)←Qe(s,a) forall s,a |
| 16: | No(s,a) ←1for all s,a |
| 17: | k←0 |
| 18: 19: | while search budget remains (k < K) do |
| 20: | Traverse the search tree with πk (Equation 1) |
| 21: | Expand new state sT and add it to the search tree |
| 22: | Evaluate maxa Qe(sT,a) and backup returns (Equation 3) |
| 23: | Set Nk+1(s,a) ← Nk(s,a) and then increment counts of visited states and actions |
| Compute estimates for Qk+1(s,a) (Equation 4) |
| 24: | k←k+1 |
| 25: | Return {Qk(so,ai)}i |
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| 1: procedure SAVE(θ) |
| 2: | while true do |
| 3: | Begin episode at s |
| 4: | while acting do |
| 5: | Estimate QmCTs(s,:) ← MCTS(s, Qθ) |
| 6: | Select a using epsilon-greedy from QMCTs(s,:) |
| 7: 8: | Execute a in environment and receive s',r |
| 9: | Add (s,a,r,s',QmCTs(s,·)) to replay buffer |
| s↑s` |
| 10: | while learning do |
| 11: | Sample minibatch of experience from the replay buffer |
| 12: | Update θ to minimize Equation 6 |
| 13: |
| 14: | procedure MCTS(so, Qθ) |
| 15: | Qo(s,a)←Qe(s,a) forall s,a |
| 16: | No(s,a) ←1for all s,a |
| 17: | k←0 |
| 18: 19: | while search budget remains (k < K) do |
| 20: | Traverse the search tree with πk (Equation 1) |
| 21: | Expand new state sT and add it to the search tree |
| 22: | Evaluate maxa Qe(sT,a) and backup returns (Equation 3) |
| 23: | Set Nk+1(s,a) ← Nk(s,a) and then increment counts of visited states and actions |
| Compute estimates for Qk+1(s,a) (Equation 4) |
| 24: | k←k+1 |
| 25: | Return {Qk(so,ai)}i |
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