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States

Variables

Logic

In this section, we accept that by accomplishing activity \$a\$ from accompaniment \$s\$, we deterministically access in accompaniment \$textrm{Succ}(s,a)\$. The ambition actuality is to actuate a arrangement of accomplishments \$(a_1,a_2,a_3,a_4,…)\$ that starts from an antecedent accompaniment and leads to an end state. In adjustment to break this affectionate of problem, our cold will be to acquisition the minimum amount aisle by application states-based models.

This class of states-based algorithms explores all accessible states and actions. It is absolutely anamnesis efficient, and is acceptable for huge accompaniment spaces but the runtime can become exponential in the affliction cases.

Search botheration A chase botheration is authentic with:

The cold is to acquisition a aisle that minimizes the cost.

Backtracking chase Backtracking chase is a aboveboard recursive algorithm that tries all possibilities to acquisition the minimum amount path. Here, activity costs can be either absolute or negative.

Breadth-first chase (BFS) Breadth-first chase is a blueprint chase algorithm that does a level-by-level traversal. We can apparatus it iteratively with the advice of a chain that food at anniversary footfall approaching nodes to be visited. For this algorithm, we can accept activity costs to be according to a connected \$cgeqslant0\$.

Depth-first chase (DFS) Depth-first chase is a chase algorithm that traverses a blueprint by afterward anniversary aisle as abysmal as it can. We can apparatus it recursively, or iteratively with the advice of a assemblage that food at anniversary footfall approaching nodes to be visited. For this algorithm, activity costs are affected to be according to 0.

Iterative deepening The accepted deepening ambush is a modification of the depth-first chase algorithm so that it stops afterwards extensive a assertive depth, which guarantees optimality back all activity costs are equal. Here, we accept that activity costs are according to a connected \$cgeqslant0\$.

Tree chase algorithms arbitrary By acquainted \$b\$ the cardinal of accomplishments per state, \$d\$ the band-aid depth, and \$D\$ the best depth, we have:

This class of states-based algorithms aims at amalgam optimal paths, enabling exponential savings. In this section, we will focus on activating programming and compatible amount search.

Graph A blueprint is comprised of a set of vertices \$V\$ (also alleged nodes) as able-bodied as a set of edges \$E\$ (also alleged links).

State A accompaniment is a arbitrary of all accomplished accomplishments acceptable to accept approaching accomplishments optimally.

Dynamic programming Activating programming (DP) is a backtracking chase algorithm with memoization (i.e. fractional after-effects are saved) whose ambition is to acquisition a minimum amount aisle from accompaniment \$s\$ to an end accompaniment \$s_textrm{end}\$. It can potentially accept exponential accumulation compared to acceptable blueprint chase algorithms, and has the acreage to alone assignment for acyclic graphs. For any accustomed accompaniment \$s\$, the approaching amount is computed as follows:

[boxed{textrm{FutureCost}(s)=left{begin{array}{lc}0 & textrm{if IsEnd}(s)\underset{aintextrm{Actions}(s)}{textrm{min}}big[textrm{Cost}(s,a) textrm{FutureCost(Succ}(s,a))big] & textrm{otherwise}end{array}right.}]

Types of states The table beneath presents the analogue back it comes to states in the ambience of compatible amount search:

Uniform amount chase Compatible amount chase (UCS) is a chase algorithm that aims at award the beeline aisle from a accompaniment \$s_textrm{start}\$ to an end accompaniment \$s_textrm{end}\$. It explores states \$s\$ in accretion adjustment of \$textrm{PastCost}(s)\$ and relies on the actuality that all activity costs are non-negative.

Correctness assumption Back a accompaniment \$s\$ is popped from the borderland \$mathcal{F}\$ and confused to explored set \$mathcal{E}\$, its antecedence is according to \$textrm{PastCost}(s)\$ which is the minimum amount aisle from \$s_textrm{start}\$ to \$s\$.

Graph chase algorithms arbitrary By acquainted \$N\$ the cardinal of absolute states, \$n\$ of which are explored afore the end accompaniment \$s_textrm{end}\$, we have:

Suppose we are not accustomed the ethics of \$textrm{Cost}(s,a)\$, we appetite to appraisal these quantities from a training set of minimizing-cost-path arrangement of accomplishments \$(a_1, a_2, …, a_k)\$.

Structured perceptron The structured perceptron is an algorithm aiming at iteratively acquirements the amount of anniversary state-action pair. At anniversary step, it:

Heuristic activity A heuristic is a activity \$h\$ over states \$s\$, area anniversary \$h(s)\$ aims at ciphering \$textrm{FutureCost}(s)\$, the amount of the aisle from \$s\$ to \$s_textrm{end}\$.

Algorithm \$A^{*}\$ is a chase algorithm that aims at award the beeline aisle from a accompaniment \$s\$ to an end accompaniment \$s_textrm{end}\$. It explores states \$s\$ in accretion adjustment of \$textrm{PastCost}(s) h(s)\$. It is agnate to a compatible amount chase with bend costs \$textrm{Cost}'(s,a)\$ accustomed by:

[boxed{textrm{Cost}'(s,a)=textrm{Cost}(s,a) h(textrm{Succ}(s,a))-h(s)}]

Consistency A heuristic \$h\$ is said to be connected if it satisfies the two afterward properties:

[boxed{h(s) leqslant textrm{Cost}(s,a) h(textrm{Succ}(s,a))}]

[boxed{h(s_{textrm{end}})=0}]

Correctness If \$h\$ is consistent, again \$A^*\$ allotment the minimum amount path.

Admissibility A heuristic \$h\$ is said to be acceptable if we have:

[boxed{h(s)leqslanttextrm{FutureCost}(s)}]

Theorem Let \$h(s)\$ be a accustomed heuristic. We have:

[boxed{h(s)textrm{ consistent}Longrightarrow h(s)textrm{ admissible}}]

Efficiency \$A^*\$ explores all states \$s\$ acceptable the afterward equation:

[boxed{textrm{PastCost}(s)leqslanttextrm{PastCost}(s_{textrm{end}})-h(s)}]

It is a framework for bearing connected heuristics. The abstraction is to acquisition closed-form bargain costs by removing constraints and use them as heuristics.

Relaxed chase botheration The alleviation of chase botheration \$P\$ with costs \$textrm{Cost}\$ is denoted \$P_{textrm{rel}}\$ with costs \$textrm{Cost}_{textrm{rel}}\$, and satisfies the identity:

[boxed{textrm{Cost}_{textrm{rel}}(s,a)leqslanttextrm{Cost}(s,a)}]

Relaxed heuristic Accustomed a airy chase botheration \$P_{textrm{rel}}\$, we ascertain the airy heuristic \$h(s)=textrm{FutureCost}_{textrm{rel}}(s)\$ as the minimum amount aisle from \$s\$ to an end accompaniment in the blueprint of costs \$textrm{Cost}_{textrm{rel}}(s,a)\$.

Consistency of airy heuristics Let \$P_{textrm{rel}}\$ be a accustomed airy problem. By theorem, we have:

[boxed{h(s)=textrm{FutureCost}_{textrm{rel}}(s)Longrightarrow h(s)textrm{ consistent}}]

Tradeoff back allotment heuristic We accept to antithesis two aspects in allotment a heuristic:

Max heuristic Let \$h_1(s)\$, \$h_2(s)\$ be two heuristics. We accept the afterward property:

[boxed{h_1(s),textrm{ }h_2(s)textrm{ consistent}Longrightarrow h(s)=max{h_1(s),textrm{ }h_2(s)}textrm{ consistent}}]

In this section, we accept that bold activity \$a\$ from accompaniment \$s\$ can advance to several states \$s_1′,s_2′,…\$ in a probabilistic manner. In adjustment to acquisition our way amid an antecedent accompaniment and an end state, our cold will be to acquisition the best amount activity by application Markov accommodation processes that advice us cope with randomness and uncertainty.

Definition The cold of a Markov accommodation activity is to aerate rewards. It is authentic with:

Transition probabilities The alteration anticipation \$T(s,a,s’)\$ specifies the anticipation of activity to accompaniment \$s’\$ afterwards activity \$a\$ is taken in accompaniment \$s\$. Anniversary \$s’ mapsto T(s,a,s’)\$ is a anticipation distribution, which agency that:

[forall s,a,quadboxed{displaystylesum_{s’intextrm{ States}}T(s,a,s’)=1}]

Policy A activity \$pi\$ is a activity that maps anniversary accompaniment \$s\$ to an activity \$a\$, i.e.

[boxed{pi : s mapsto a}]

Utility The annual of a aisle \$(s_0, …, s_k)\$ is the discounted sum of the rewards on that path. In added words,

[boxed{u(s_0,…,s_k)=sum_{i=1}^{k}r_igamma^{i-1}}]

The amount aloft is an analogy of the case \$k=4\$.

Q-value The \$Q\$-value of a activity \$pi\$ at accompaniment \$s\$ with activity \$a\$, additionally denoted \$Q_{pi}(s,a)\$, is the accepted annual from accompaniment \$s\$ afterwards demography activity \$a\$ and again afterward activity \$pi\$. It is authentic as follows:

[boxed{Q_{pi}(s,a)=sum_{s’intextrm{ States}}T(s,a,s’)left[textrm{Reward}(s,a,s’) gamma V_pi(s’)right]}]

Value of a activity The amount of a activity \$pi\$ from accompaniment \$s\$, additionally denoted \$V_{pi}(s)\$, is the accepted annual by afterward activity \$pi\$ from accompaniment \$s\$ over accidental paths. It is authentic as follows:

[boxed{V_pi(s)=Q_pi(s,pi(s))}]

Policy appraisal Accustomed a activity \$pi\$, activity appraisal is an accepted algorithm that aims at ciphering \$V_pi\$. It is done as follows:

[boxed{V_pi^{(0)}(s)longleftarrow0}]

[forall s,quadboxed{V_pi^{(t)}(s)longleftarrow Q_pi^{(t-1)}(s,pi(s))}]

[boxed{Q_pi^{(t-1)}(s,pi(s))=sum_{s’intextrm{ States}}T(s,pi(s),s’)Big[textrm{Reward}(s,pi(s),s’) gamma V_pi^{(t-1)}(s’)Big]}]

Optimal Q-value The optimal \$Q\$-value \$Q_{textrm{opt}}(s,a)\$ of accompaniment \$s\$ with activity \$a\$ is authentic to be the best \$Q\$-value accomplished by any policy. It is computed as follows:

[boxed{Q_{textrm{opt}}(s,a)=sum_{s’intextrm{ States}}T(s,a,s’)left[textrm{Reward}(s,a,s’) gamma V_textrm{opt}(s’)right]}]

Optimal amount The optimal amount \$V_{textrm{opt}}(s)\$ of accompaniment \$s\$ is authentic as actuality the best amount accomplished by any policy. It is computed as follows:

[boxed{V_{textrm{opt}}(s)=underset{aintextrm{ Actions}(s)}{textrm{max}}Q_textrm{opt}(s,a)}]

Optimal activity The optimal activity \$pi_{textrm{opt}}\$ is authentic as actuality the activity that leads to the optimal values. It is authentic by:

[forall s,quadboxed{pi_{textrm{opt}}(s)=underset{aintextrm{ Actions}(s)}{textrm{argmax}}Q_textrm{opt}(s,a)}]

Value abundance Amount abundance is an algorithm that finds the optimal amount \$V_{textrm{opt}}\$ as able-bodied as the optimal activity \$pi_{textrm{opt}}\$. It is done as follows:

[boxed{V_{textrm{opt}}^{(0)}(s)longleftarrow0}]

[forall s,quadboxed{V_textrm{opt}^{(t)}(s)longleftarrow underset{aintextrm{ Actions}(s)}{textrm{max}}Q_textrm{opt}^{(t-1)}(s,a)}]

[boxed{Q_textrm{opt}^{(t-1)}(s,a)=sum_{s’intextrm{ States}}T(s,a,s’)Big[textrm{Reward}(s,a,s’) gamma V_textrm{opt}^{(t-1)}(s’)Big]}]

Now, let’s accept that the alteration probabilities and the rewards are unknown.

Model-based Monte Carlo The model-based Monte Carlo adjustment aims at ciphering \$T(s,a,s’)\$ and \$textrm{Reward}(s,a,s’)\$ application Monte Carlo simulation with:

[boxed{widehat{T}(s,a,s’)=frac{#textrm{ times }(s,a,s’)textrm{ occurs}}{#textrm{ times }(s,a)textrm{ occurs}}}]

[boxed{widehat{textrm{Reward}}(s,a,s’)=rtextrm{ in }(s,a,r,s’)}]

Model-free Monte Carlo The model-free Monte Carlo adjustment aims at anon ciphering \$Q_{pi}\$, as follows:

[boxed{widehat{Q}_pi(s,a)=textrm{average of }u_ttextrm{ area }s_{t-1}=s, a_t=a}]

Equivalent conception By introducing the connected \$eta=frac{1}{1 (#textrm{updates to }(s,a))}\$ and for anniversary \$(s,a,u)\$ of the training set, the amend aphorism of model-free Monte Carlo has a arched aggregate formulation:

[boxed{widehat{Q}_pi(s,a)leftarrow(1-eta)widehat{Q}_pi(s,a) eta u}]

[boxed{widehat{Q}_pi(s,a)leftarrowwidehat{Q}_pi(s,a) – eta (widehat{Q}_pi(s,a) – u)}]

SARSA State-action-reward-state-action (SARSA) is a boostrapping adjustment ciphering \$Q_pi\$ by application both raw abstracts and estimates as allotment of the amend rule. For anniversary \$(s,a,r,s’,a’)\$, we have:

[boxed{widehat{Q}_pi(s,a)longleftarrow(1-eta)widehat{Q}_pi(s,a) etaBig[r gammawidehat{Q}_pi(s’,a’)Big]}]

Q-learning \$Q\$-learning is an off-policy algorithm that produces an appraisal for \$Q_textrm{opt}\$. On anniversary \$(s,a,r,s’,a’)\$, we have:

[boxed{widehat{Q}_{textrm{opt}}(s,a)leftarrow(1-eta)widehat{Q}_{textrm{opt}}(s,a) etaBig[r gammaunderset{a’intextrm{ Actions}(s’)}{textrm{max}}widehat{Q}_{textrm{opt}}(s’,a’)Big]}]

Epsilon-greedy The epsilon-greedy activity is an algorithm that balances analysis with anticipation \$epsilon\$ and corruption with anticipation \$1-epsilon\$. For a accustomed accompaniment \$s\$, the activity \$pi_{textrm{act}}\$ is computed as follows:

[boxed{pi_textrm{act}(s)=left{begin{array}{ll}underset{aintextrm{ Actions}}{textrm{argmax }}widehat{Q}_textrm{opt}(s,a) & textrm{with proba }1-epsilon\textrm{random from Actions}(s) & textrm{with proba }epsilonend{array}right.}]

In amateur (e.g. chess, backgammon, Go), added agents are present and charge to be taken into annual back amalgam our policy.

Game timberline A bold timberline is a timberline that describes the possibilities of a game. In particular, anniversary bulge is a accommodation point for a amateur and anniversary root-to-leaf aisle is a accessible aftereffect of the game.

Two-player zero-sum bold It is a bold area anniversary accompaniment is absolutely empiric and such that players booty turns. It is authentic with:

Types of behavior There are two types of policies:

Expectimax For a accustomed accompaniment \$s\$, the expectimax amount \$V_{textrm{exptmax}}(s)\$ is the best accepted annual of any abettor activity back arena with account to a anchored and accepted antagonist activity \$pi_{textrm{opp}}\$. It is computed as follows:

[boxed{V_{textrm{exptmax}}(s)=left{begin{array}{ll}textrm{Utility}(s) & textrm{IsEnd}(s)\underset{aintextrm{Actions}(s)}{textrm{max}}V_{textrm{exptmax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{agent}\displaystylesum_{aintextrm{Actions}(s)}pi_{textrm{opp}}(s,a)V_{textrm{exptmax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{opp}end{array}right.}]

Minimax The ambition of minimax behavior is to acquisition an optimal activity adjoin an antagonist by bold the affliction case, i.e. that the antagonist is accomplishing aggregate to abbreviate the agent’s utility. It is done as follows:

[boxed{V_{textrm{minimax}}(s)=left{begin{array}{ll}textrm{Utility}(s) & textrm{IsEnd}(s)\underset{aintextrm{Actions}(s)}{textrm{max}}V_{textrm{minimax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{agent}\underset{aintextrm{Actions}(s)}{textrm{min}}V_{textrm{minimax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{opp}end{array}right.}]

Minimax backdrop By acquainted \$V\$ the amount function, there are 3 backdrop about minimax to accept in mind:

[boxed{forall pi_{textrm{agent}},quad V(pi_{textrm{max}},pi_{textrm{min}})geqslant V(pi_{textrm{agent}},pi_{textrm{min}})}]

[boxed{forall pi_{textrm{opp}},quad V(pi_{textrm{max}},pi_{textrm{min}})leqslant V(pi_{textrm{max}},pi_{textrm{opp}})}]

[boxed{forall pi,quad V(pi_{textrm{max}},pi)leqslant V(pi_{textrm{exptmax}},pi)}]

[boxed{V(pi_{textrm{exptmax}},pi_{textrm{min}})leqslant V(pi_{textrm{max}},pi_{textrm{min}})leqslant V(pi_{textrm{max}},pi)leqslant V(pi_{textrm{exptmax}},pi)}]

Evaluation activity An appraisal activity is a domain-specific and almost appraisal of the amount \$V_{textrm{minimax}}(s)\$. It is denoted \$textrm{Eval}(s)\$.

Alpha-beta pruning Alpha-beta pruning is a domain-general exact adjustment optimizing the minimax algorithm by alienated the accidental analysis of genitalia of the bold tree. To do so, anniversary amateur keeps clue of the best amount they can achievement for (stored in \$alpha\$ for the maximizing amateur and in \$beta\$ for the aspersing player). At a accustomed step, the activity \$beta < alpha\$ agency that the optimal aisle is not activity to be in the accepted annex as the beforehand amateur had a bigger advantage at their disposal.

TD acquirements Temporal aberration (TD) acquirements is acclimated back we don’t apperceive the transitions/rewards. The amount is based on analysis policy. To be able to use it, we charge to apperceive rules of the bold \$textrm{Succ}(s,a)\$. For anniversary \$(s,a,r,s’)\$, the amend is done as follows:

[boxed{wlongleftarrow w-etabig[V(s,w)-(r gamma V(s’,w))big]nabla_wV(s,w)}]

This is the adverse of turn-based games, area there is no acclimation on the player’s moves.

Single-move accompanying bold Let there be two players \$A\$ and \$B\$, with accustomed accessible actions. We agenda \$V(a,b)\$ to be \$A\$’s annual if \$A\$ chooses activity \$a\$, \$B\$ chooses activity \$b\$. \$V\$ is alleged the adjustment matrix.

Strategies There are two capital types of strategies:

[boxed{aintextrm{Actions}}]

[forall aintextrm{Actions},quadboxed{0leqslantpi(a)leqslant1}]

Game appraisal The amount of the bold \$V(pi_A,pi_B)\$ back amateur \$A\$ follows \$pi_A\$ and amateur \$B\$ follows \$pi_B\$ is such that:

[boxed{V(pi_A,pi_B)=sum_{a,b}pi_A(a)pi_B(b)V(a,b)}]

Minimax assumption By acquainted \$pi_A,pi_B\$ alignment over alloyed strategies, for every accompanying two-player zero-sum bold with a bound cardinal of actions, we have:

[boxed{max_{pi_A}min_{pi_B}V(pi_A,pi_B)=min_{pi_B}max_{pi_A}V(pi_A,pi_B)}]

Payoff cast We ascertain \$V_p(pi_A,pi_B)\$ to be the annual for amateur \$p\$.

Nash calm A Nash calm is \$(pi_A^*,pi_B^*)\$ such that no amateur has an allurement to change its strategy. We have:

[boxed{forall pi_A, V_A(pi_A^*,pi_B^*)geqslant V_A(pi_A,pi_B^*)}quadtextrm{and}quadboxed{forall pi_B, V_B(pi_A^*,pi_B^*)geqslant V_B(pi_A^*,pi_B)}]

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