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href="https://github.com/JuliaDecisionFocusedLearning/DecisionFocusedLearningBenchmarks.jl/blob/main/docs/src/benchmarks/contextual_stochastic_argmax.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Contextual-Stochastic-Argmax"><a class="docs-heading-anchor" href="#Contextual-Stochastic-Argmax">Contextual Stochastic Argmax</a><a id="Contextual-Stochastic-Argmax-1"></a><a class="docs-heading-anchor-permalink" href="#Contextual-Stochastic-Argmax" title="Permalink"></a></h1><p><a href="../../api/#DecisionFocusedLearningBenchmarks.ContextualStochasticArgmax.ContextualStochasticArgmaxBenchmark"><code>ContextualStochasticArgmaxBenchmark</code></a> is a minimalist contextual stochastic optimization benchmark problem.</p><p>The decision maker selects one item out of <span>$n$</span>. Item values are uncertain at decision time: they depend on a base utility plus a context-correlated perturbation revealed only after the decision is made. An observable context vector, correlated with the perturbation via a fixed linear map <span>$W$</span>, allows the learner to anticipate the perturbation and pick the right item.</p><h2 id="Problem-Formulation"><a class="docs-heading-anchor" href="#Problem-Formulation">Problem Formulation</a><a id="Problem-Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-Formulation" title="Permalink"></a></h2><p><strong>Instance</strong>: <span>$c_{\text{base}} \sim \mathcal{U}[0,1]^n$</span>, base values for <span>$n$</span> items.</p><p><strong>Context</strong>: <span>$x_{\text{raw}} \sim \mathcal{N}(0, I_d)$</span>, a <span>$d$</span>-dimensional signal correlated with item values. The feature vector passed to the model is <span>$x = [c_{\text{base}};\, x_{\text{raw}}] \in \mathbb{R}^{n+d}$</span>.</p><p><strong>Scenario</strong>: the realized item values are</p><p class="math-container">\[\xi = c_{\text{base}} + W x_{\text{raw}} + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I_n)\]</p><p>where <span>$W \in \mathbb{R}^{n \times d}$</span> is a fixed matrix unknown to the learner.</p><p><strong>Decision</strong>: <span>$y \in \{e_1, \ldots, e_n\}$</span> (one-hot vector selecting one item).</p><h2 id="Policies"><a class="docs-heading-anchor" href="#Policies">Policies</a><a id="Policies-1"></a><a class="docs-heading-anchor-permalink" href="#Policies" title="Permalink"></a></h2><h3 id="DFL-Policy"><a class="docs-heading-anchor" href="#DFL-Policy">DFL Policy</a><a id="DFL-Policy-1"></a><a class="docs-heading-anchor-permalink" href="#DFL-Policy" title="Permalink"></a></h3><p class="math-container">\[\xrightarrow[\text{Features}]{x}
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