β-Optimization in the Information Bottleneck Framework: A Theoretical Analysis

The Information Bottleneck (IB) framework formalizes the trade-off between compression and prediction in representation learning. A crucial parameter is the Lagrange multiplier β, which controls the balance between preserving information relevant to a target variable Y and compressing the representation Z of an input X. Selecting an optimal β (denoted β∗) is challenging and typically done via empirical tuning. In this paper, I present a rigorous theoretical analysis of β∗-optimization in both the Variational IB (VIB) and Neural IB (NIB) settings. I define β∗ as the critical value of β that marks the boundary between non-trivial (informative) and trivial (uninformative) representations, ensuring maximal compression before the representation collapses. I derive formal conditions for its existence and uniqueness. I prove several key results: (1) the IB trade-off curve (relevance–compression frontier) is concave under mild conditions, implying that β, as the slope of this curve, uniquely characterizes optimal operating points in regular cases; (2) there exists a critical β threshold, β∗ = F′(0+) (the slope of the IB curve at zero compression), beyond which the IB solution collapses to a trivial representation; (3) for practical IB implementations (VIB and NIB), I discuss how β∗ can be computed algorithmically, including complexity analysis of naive β-sweeping versus adaptive methods like binary search, for which pseudo-code is provided. I provide formal theorems and proofs for concavity properties of the IB Lagrangian, continuity of the IB curve, and boundedness of mutual information quantities. Furthermore, I compare standard IB, VIB, and NIB formulations in terms of the optimal β, showing that while standard IB provides a theoretical target for β∗, variational and neural approximations may deviate from this optimum. My analysis is complemented by a discussion on the implications for deep neural network representations. The results establish a principled foundation for β selection in IB, guiding practitioners to achieve maximal meaningful compression without exhaustive trial-and-error