Characterizing Learning in Deep Neural Networks using Tractable Algorithmic Complexity Analysis

Why does weight decay work? We prove that, in any fixed-precision regime, the smallest weight norm of a looped neural network outputting a binary string equals the Kolmogorov complexity of that string, up to a logarithmic factor. This implies that weight decay induces a prior matching Solomonoff's universal prior, the optimal prior over computable functions, up to a polynomial factor. The result is norm-agnostic: in fixed precision, every weight norm collapses to the non-zero parameter count up to constants, so the same sandwich bound holds for any norm used as a regulariser. The proof has two short reductions: any program for a universal Turing machine can be encoded into neural weights at unit cost per program bit, and any fixed-precision network can be described by enumerating its non-zero parameters with logarithmic addressing overhead. Both bounds are tight up to constants, with the logarithmic factor realised by permutation encodings: a network whose parameters encode a permutation produces a string whose Kolmogorov complexity is the non-zero parameter count times its logarithm. The fixed-precision assumption is essential: with infinite precision, neural networks can encode non-computable functions and the weight norm loses its relevance.

We present HiReLC, a hierarchical ensemble-reinforcement learning framework for automated joint quantization and structured pruning of deep neural networks. The framework decomposes the compression search across two levels of abstraction: low-level agents (LLAs) operate independently per block, selecting per-kernel configurations over a multi-discrete action space spanning bitwidth, pruning keep-ratio, quantization type, and granularity, while high-level agents (HLAs) coordinate global budget allocation via ensemble voting guided by Fisher Information-based sensitivity estimates. To mitigate the computational cost of policy evaluation, an iterative active learning loop interleaves surrogate-guided RL optimization with post-compression fine-tuning, using a lightweight MLP surrogate to amortize expensive evaluations and a logit-MSE proxy during cold-start. The surrogate is used for reward shaping rather than as a replacement for final post-compression evaluation. The controller is architecture-agnostic by design, with a modular layer abstraction decoupling the RL environment from the underlying network topology. Experiments across Vision Transformer and CNN benchmarks demonstrate effective parameter-storage compression ratios of 5.99 - 6.72$\times$ with a 3.83 % gain in one setting and 0.55 - 5.62 % accuracy drops elsewhere, supporting hierarchical policy decomposition and sensitivity-aware guidance as practical design choices for joint neural network compression.

Algorithmic Information Dynamics (AID) studies systems by perturbing them and measuring changes in algorithmic complexity, but its usual estimator, the Block Decomposition Method, is piecewise constant, restricting the calculus to finite differences. We use $K^{\mathrm{CDM}}_{\mathrm{s}F}$, a certified, differentiable estimator, to bring the calculus into learning dynamics: grokking, where a complexity order parameter is known but has not been made to act. As a transient loss kick, the estimator becomes a controller that accelerates grokking in Levin's description-length--versus-time sense, within a data-dependent Occam boundary whose finite-size trend, $f_c\sim\ln p/p$, is consistent with a coupon-collector interpretation. Ablations show that a complexity gate matches a train-loss gate in rescuing failing seeds with $27\%$ less intervention; among the tested signals, only map complexity marks the transition's completion; the certified prior and the per-parameter $\nabla K$ attribution are both fungible (a uniform-prior sensor makes bit-identical gate decisions, and random supports match $\nabla K$-selected ones above a sparsity threshold); and direct field perturbation shows a nucleation-like response to the Occam field (no linear regime is resolved over the probed amplitudes, so these measurements do not justify a fluctuation--dissipation surrogate), with a finite-field response growing by orders of magnitude toward the phase-transition. These measurements account for the empirically tuned staircase: bang--bang pulses, stall-fired and released on yield, whose iteration plausibly builds the response it exploits. The kick transfers to sparse parity and to a transformer; a sustained weight-space loss fails. The algorithmic estimator's distinct contribution is timing (when to fire and when to release), not attribution.