From 6ab8833fb4b2489a269a7968411f0151b3e5b881 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Thu, 25 Sep 2025 11:22:31 -0500 Subject: [PATCH] updates --- content/CSE510/CSE510_L10.md | 280 ++++++++++++++++++ content/CSE510/_meta.js | 1 + .../Math401/Extending_thesis/Math401_S3.md | 1 + .../Maximization_bias_of_Q-learning.png | Bin 0 -> 92109 bytes 4 files changed, 282 insertions(+) create mode 100644 content/CSE510/CSE510_L10.md create mode 100644 content/Math401/Extending_thesis/Math401_S3.md create mode 100644 public/CSE510/Maximization_bias_of_Q-learning.png diff --git a/content/CSE510/CSE510_L10.md b/content/CSE510/CSE510_L10.md new file mode 100644 index 0000000..9d982f2 --- /dev/null +++ b/content/CSE510/CSE510_L10.md @@ -0,0 +1,280 @@ +# CSE510 Deep Reinforcement Learning (Lecture 10) + +## Deep Q-network (DQN) + +Network input = Observation history + +- Window of previous screen shots in Atari + +Network output = One output node per action (returns Q-value) + +### Stability issues of DQN + +Naïve Q-learning oscillates or diverges with neural nets + +Data is sequential and successive samples are correlated (time-correlated) + +- Correlations present in the sequence of observations +- Correlations between the estimated value and the target values +- Forget previous experiences and overfit similar correlated samples + +Policy changes rapidly with slight changes to Q-values + +- Policy may oscillate +- Distribution of data can swing from one extreme to another + +Scale of rewards and Q-values is unknown + +- Gradients can be unstable when back-propagated + +### Deadly Triad in Reinforcement Learning + +Off-policy learning (learning the expected reward changes of policy change instead of the optimal policy) +Function approximation (usually with supervised learning) + +$Q(s,a)\gets f_\theta(s,a)$ + +Bootstrapping (self-reference) + +- $Q(s,a)\gets r(s,a)+\gamma \max_{a'\in A} Q(s',a')$ + +### Stable Solutions for DQN + +DQN provides a stable solution to deep value-based RL + +1. Experience replay +2. Freeze target Q-network +3. Clip rewards to sensible range + +#### Experience replay + +To remove correlations, build dataset from agent's experience + +- Take action $a_t$ +- Store transition $(s_t, a_t, r_t, s_{t+1})$ in replay memory $D$ +- Sample random mini-batch of transitions $(s,a,r,s')$ from replay memory $D$ +- Optimize Mean Squared Error between Q-network and Q-learning target + +$$ +L_i(\theta_i) = \mathbb{E}_{(s,a,r,s') \sim U(D)} \left[ \left( r+\gamma \max_{a'\in A} Q(s',a';\theta_i^-)-Q(s,a;\theta_i) \right)^2 \right] +$$ + +Here $U(D)$ is the uniform distribution over the replay memory $D$. + +#### Fixed Target Q-Network + +To avoid oscillations, fix parameters used in Q- learning target + +- Compute Q-learning target w.r.t old, fixed parameters +- Optimize MSE between Q-learning targets and Q-network +- Periodically update target Q-network parameters + +#### Reward/Value Range + +- To limit impact of any one update, control the reward / value range +- DQN clips the rewards to $[-1, +1]$ + - Prevents too large Q-values + - Ensures gradients are well-conditioned + +### DQN Implementation + +#### Preprocessing + +- Raw images: $210\times 160$ pixel images with 128-color palette +- Rescaled images: $84\times 84$ +- Input: $84\times 84\times 4$ (4 most recent frames) + +#### Training + +DQN source code: +sites.google.com/a/deepmind.com/ + +- 49 Atari 2600 games +- Use RMSProp algorithms with minibatches 32 +- Use 50 million frames (38 days) +- Replay memory contains 1 million recent frames +- Agent select actions on every 4th frames + +#### Evaluation + +- Agent plays each games 30 times for 5 min with random initial conditions +- Human plays the games in the same scenarios +- Random agent play in the same scenarios to obtain baseline performance + +### DeepMind Atari + +Beat human players in 49 out of 49 games + +Strengths: + +- Quick-moving, short-horizon games +- Pinball (2539%) + +Weakness: + +- Long-horizon games that do not converge +- Walk-around games +- Montezuma’s revenge + +### DQN Summary + +- Deep Q-network agent can learn successful policies directly from high-dimensional input using end-to-end reinforcement learning + +- The algorithm achieve a level surpassing professional human games tester across 49 games + +## Extensions of DQN + +- Double Q-learning for fighting maximization bias +- Prioritized experience replay +- Dueling Q networks +- Multistep returns +- Distributed DQN + +### Double Q-learning for fighting maximization bias + +#### Maximization Bias for Q-learning + +![Maximization Bias of Q-learning](https://notenextra.trance-0.com/CSE510/Maximization_bias_of_Q-learning.png) + +False signals from $\mathcal{N}(0.1,1)$, may have few positive results from random noise. (However, in the long run, it will converge to the expected negative value.) + +#### Double Q-learning + +(Hado van Hasselt 2010) + +Train 2 action-value functions, Q1 and Q2 + +Do Q-learning on both, but + +- never on the same time steps (Q1 and Q2 are indep.) +- pick Q1 or Q2 at random to be updated on each step + +If updating Q1, use Q2 for the value of the next state: + +$$ +Q_1(S_t,A_t) \gets Q_1(S_t,A_t) + \alpha (R_{t+1} + \gamma Q_2(S_{t+1}, \arg\max_{a'\in A} Q_1(S_{t+1},a')) - Q_1(S_t,A_t)) +$$ + +Action selections are (say) $\epsilon$-greedy with respect to the sum of Q1 and Q2. (unbiased estimation and same convergence as Q-learning) + +Drawbacks: + +- More computationally expensive (only one function is trained at a time) + +```pseudocode +Initialize Q1 and Q2 +For each episode: + Initialize state + For each step: + Choose $A$ from $S$ using policy derived from Q1 and Q2 + Take action $A$, observe $R$ and $S'$ + With probability $0.5$, update Q1: + $Q1(S,A) \gets Q1(S,A) + \alpha (R + \gamma Q2(S', \arg\max_{a'\in A} Q1(S',a')) - Q1(S,A))$ + Otherwise, update Q2: + $Q2(S,A) \gets Q2(S,A) + \alpha (R + \gamma Q1(S', \arg\max_{a'\in A} Q2(S',a')) - Q2(S,A))$ + $S \gets S'$ + End for +End for +``` + +#### Double DQN + +(van Hasselt, Guez, Silver, 2015) + +A better implementation of Double Q-learning. + +- Dealing with maximization bias of Q-Learning +- Current Q-network $w$ is used to select actions +- Older Q-network $w^-$ is used to evaluate actions + +$$ +l=\left(r+\gamma Q(s', \arg\max_{a'\in A} Q(s',a';w);w^-) - Q(s,a;w)\right)^2 +$$ + +Here $\arg\max_{a'\in A} Q(s',a';w)$ is the action selected by the current Q-network $w$. + +$Q(s', \arg\max_{a'\in A} Q(s',a';w);w^-)$ is the action evaluation by the older Q-network $w^-$. + +### Prioritized Experience Replay + +(Schaul, Quan, Antonoglou, Silver, ICLR 2016) + +Weight experience according to "surprise" (or error) + +- Store experience in priority queue according to DQN error + $$ + \left|r+\gamma \arg\max_{a'\in A} Q(s',a',w^-)-Q(s,a,w)\right| + $$ + +- Stochastic Prioritization + $$ + P(i)=\frac{p_i^\alpha}{\sum_k p_k^\alpha} + $$ + - $p_i$ is proportional to the DQN error + +- $\alpha$ determines how much prioritization is used, with $\alpha = 0$ corresponding to the uniform case. + +### Dueling Q networks + +(Wang et.al., ICML, 2016) + +- Split Q-network into two channels + + - Action-independent value function $V(s; w)$: measures how good is the state $s$ + + - Action-dependent advantage function $A(s, a; w)$: measure how much better is action $a$ than the average action in state $s$ + $$ + Q(s,a; w) = V(s; w) + A(s, a; w) + $$ + +- Advantage function is defined as: + $$ + A^\pi(s, a) = Q^\pi(s, a) - V^\pi(s) + $$ + +The value stream learns to pay attention to the road + +**The advantage stream**: pay attention only when there are cars immediately in front, so as to avoid collisions + +### Multistep returns + +Truncated n-step return from a state $s_t$ + +$$ +R^{n}_t = \sum_{i=0}^{n-1} \gamma^{(k)}_t R_{t+k+1} +$$ + +Multistep Q-learning update rule: + +$$ +I=\left(R^{n}_t + \gamma^{(n)}_t \max_{a'\in A} Q(s_{t+n},a';w)-Q(s,a,w)\right)^2 +$$ + +Singlestep Q-learning update rule: + +$$ +I=\left(r+\gamma \max_{a'\in A} Q(s',a';w)-Q(s,a,w)\right)^2 +$$ + +### Distributed DQN + +- Separating Learning from Acting +- Distributing hundreds of actors over CPUs +- Advantages: better harnessing computation, local priority evaluation, better exploration + +#### Distributed DQN with Recurrent Experience Replay (R2D2) + +Providing an LSTM layer after the convolutional stack + +- To deal with partial observability + +Other tricks: + +- prioritized distributed replay +- n-step double Q-learning (with n = 5) +- generating experience by a large number of actors (typically 256) +- learning from batches of replayed experience by a single learner + +#### Agent 57 + +[link to paper](https://deepmind.google/discover/blog/agent57-outperforming-the-human-atari-benchmark/) \ No newline at end of file diff --git a/content/CSE510/_meta.js b/content/CSE510/_meta.js index 2eee658..3a5efda 100644 --- a/content/CSE510/_meta.js +++ b/content/CSE510/_meta.js @@ -12,4 +12,5 @@ export default { CSE510_L7: "CSE510 Deep Reinforcement Learning (Lecture 7)", CSE510_L8: "CSE510 Deep Reinforcement Learning (Lecture 8)", CSE510_L9: "CSE510 Deep Reinforcement Learning (Lecture 9)", + CSE510_L10: "CSE510 Deep Reinforcement Learning (Lecture 10)", } \ No newline at end of file diff --git a/content/Math401/Extending_thesis/Math401_S3.md 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