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Decision-focused learning (DFL) is a paradigm that integrates machine learning prediction with combinatorial optimization to make better decisions under uncertainty. Unlike traditional "predict-then-optimize" approaches that optimize prediction accuracy independently of downstream decision quality, DFL directly optimizes end-to-end decision performance.
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A typical DFL algorithm involves training a parametrized policy that combines a statistical predictor with an optimization component:
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```math
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\xrightarrow[\text{Instance}]{x}
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\fbox{Statistical model ``\varphi_w``}
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\xrightarrow[\text{Parameters}]{\theta}
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\fbox{CO algorithm ``f``}
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\xrightarrow[\text{Solution}]{y}
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```
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$$x \xrightarrow[\text{Instance}]{} \text{Statistical model } \varphi_w \xrightarrow[\text{Parameters}]{\theta} \text{CO algorithm } f \xrightarrow[\text{Solution}]{y}$$
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Where:
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-**Instance** $x$: input data (e.g., features, context)
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