Benjamin Prada

Shion Matsumoto, Raul Castillo, Benjamin Prada et al.

The forward and reverse Kullback-Leibler (KL) divergences arise as limiting objectives in learning and inference yet induce markedly different inductive biases that cannot be explained at the level of expectations alone. In this work, we introduce the Surprisal-Rényi Free Energy (SRFE), a log-moment-based functional of the likelihood ratio that lies outside the class of $f$-divergences. We show that SRFE recovers forward and reverse KL divergences as singular endpoint limits and derive local expansions around both limits in which the variance of the log-likelihood ratio appears as a first-order correction. This reveals an explicit mean-variance tradeoff governing departures from KL-dominated regimes. We further establish a Gibbs-type variational characterization of SRFE as the unique minimizer of a weighted sum of KL divergences and prove that SRFE directly controls large deviations of excess code-length via Chernoff-type bounds, yielding a precise Minimum Description Length interpretation. Together, these results identify SRFE as a variance- and tail-sensitive free-energy functional that clarifies the geometric and large-deviation structure underlying forward and reverse KL limits, without unifying or subsuming distinct learning frameworks.

Benjamin Prada, Ankur Mali

Classical circuit complexity characterizes parallel computation in purely combinatorial terms, ignoring the physical constraints that govern real hardware. The standard classes $\mathbf{NC}$, $\mathbf{AC}$, and $\mathbf{TC}$ treat unlimited fan-in, free interconnection, and polynomial gate counts as feasible -- assumptions that conflict with geometric, energetic, and thermodynamic realities. We introduce the family of realizable circuit classes $\mathbf{RC}_d$, which model computation embedded in physical $d$-dimensional space. Each circuit in $\mathbf{RC}_d$ obeys conservative realizability laws: volume scales as $\mathcal{O}(t^d)$, cross-boundary information flux is bounded by $\mathcal{O}(t^{d-1})$ per unit time, and growth occurs through local, physically constructible edits. These bounds apply to all causal systems, classical or quantum. Within this framework, we show that algorithms with runtime $ω(n^{d/(d-1)})$ cannot scale to inputs of maximal entropy, and that any $d$-dimensional parallel implementation offers at most a polynomial speed-up of degree $(d-1)$ over its optimal sequential counterpart. In the limit $d\to\infty$, $\mathbf{RC}_\infty(\mathrm{polylog})=\mathbf{NC}$, recovering classical parallelism as a non-physical idealization. By unifying geometry, causality, and information flow, $\mathbf{RC}_d$ extends circuit complexity into the physical domain, revealing universal scaling laws for computation.

Benjamin Prada, Shion Matsumoto, Abdul Malik Zekri et al.

We present the first theoretical framework that connects predictive coding (PC), a biologically inspired local learning rule, with the minimum description length (MDL) principle in deep networks. We prove that layerwise PC performs block-coordinate descent on the MDL two-part code objective, thereby jointly minimizing empirical risk and model complexity. Using Hoeffding's inequality and a prefix-code prior, we derive a novel generalization bound of the form $R(θ) \le \hat{R}(θ) + \frac{L(θ)}{N}$, capturing the tradeoff between fit and compression. We further prove that each PC sweep monotonically decreases the empirical two-part codelength, yielding tighter high-probability risk bounds than unconstrained gradient descent. Finally, we show that repeated PC updates converge to a block-coordinate stationary point, providing an approximate MDL-optimal solution. To our knowledge, this is the first result offering formal generalization and convergence guarantees for PC-trained deep models, positioning PC as a theoretically grounded and biologically plausible alternative to backpropagation.