RDCO: Rate-Distortion-Complexity Optimization

Updated 3 July 2026

RDCO is a framework that integrates rate, distortion, and computational complexity to jointly optimize codec performance across various communication systems.
It employs explicit Lagrangian formulations, scalable algorithms, and learned transforms to balance trade-offs in neural codecs, video streaming, and semantic communications.
RDCO advances practical codec design by delivering measurable gains in BD-rate, PSNR, and decoding time, addressing real-world hardware and latency constraints.

Rate-Distortion-Complexity Optimization (RDCO) is the study and engineering of systems that jointly minimize communication or storage rate and output distortion, while explicitly quantifying and constraining computational complexity. RDCO extends classical rate-distortion (RD) theory by incorporating a third axis—algorithmic or implementation complexity—transforming source coding, compression, and communication system design into a multidimensional trade-off problem. Motivation arises in modern systems where encoder/decoder speed, hardware constraints, or model complexity are actual deployment bottlenecks, such as neural codecs, semantic communication, or real-time adaptive streaming. Recent work operationalizes RDCO using explicit Lagrangian formulations, constraint-driven workflows, learned transforms/compression systems, and scalable algorithms across video, image, and semantic data domains.

1. Core Principles and Mathematical Formulation

Classically, rate-distortion theory considers minimizing channel or storage rate $R$ for a target distortion $D$ , i.e.,

$R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$

RDCO generalizes this by introducing a quantification of complexity $C$ , leading to optimization formulations such as:

$\min \; D + \lambda R + \gamma C$

where $D$ is distortion, $R$ is rate (e.g., bits/s, bpp), $C$ is complexity (e.g., runtime, FLOPs, coding time), and $\lambda, \gamma$ are tradeoff weights determined by application requirements (Queiroz et al., 7 Sep 2025). Constraints can also be imposed, e.g.,

$\min \sum_{i=1}^{M} d_i \quad \text{s.t.} \quad \sum_{i=1}^{M} r_i \le R, \; \sum_{i=1}^{M} t_i \le T,$

where $D$ 0 represent per-shot or per-block rate, distortion, and complexity (Zhong et al., 2021).

Complexity quantification must be operational and comparable for optimization: measures include single-thread CPU time (Zhong et al., 2021), number of multiply–accumulate operations (MACs) (Queiroz et al., 7 Sep 2025), decoding time (Gao et al., 2023), or mutual information-based model complexity $D$ 1 (Chai et al., 16 Feb 2026).

2. RDCO in Video Coding and Adaptive Streaming

In empirical video systems, RDCO is operationalized by expanding standard encoding pipelines:

Per-shot video coding: The parameter space is extended from (resolution, QP/CRF) pairs to $D$ 2, capturing a 3D surface $D$ 3 over rate, distortion, and encoding complexity. Convex-hull filtering, 3D model fitting (e.g., $D$ 4), and Lagrangian selection produce dense sets of operating points with explicit tradeoffs. This method enables selection of representations spanning complexities from 100% down to 3% of the slowest anchor, achieving up to 19.17% BD-rate gain at matched complexity (Zhong et al., 2021).
Corpus-level optimization: For large-scale datasets, videos are clustered by RD behavior using cheaply calculated complexity features; each cluster receives its own optimal encoder setting. Machine-learning models (such as SVMs) are trained to predict cluster membership, guiding bitrate allocation that yields 22% average BD-rate reduction while keeping average and worst-case distortion bounded (John et al., 2020).

These workflows reflect a transition from purely RD-constrained systems to joint RDCO-optimized systems, capable of meeting tight latency or compute budgets per sequence, per application, or at scale.

3. RDCO in Neural and Learned Compression

The rise of model-based codecs introduces computation cost as a critical factor:

Neural image codecs: In autoregressive models, decoding complexity is largely due to serial entropy decoding operations. RDCO is made explicit via the addition of complexity terms to the loss function, e.g., $D$ 5, where $D$ 6 quantifies complexity as the fraction of latents requiring expensive context modeling. Variable-complexity codecs train with a spatial mask that selects where to apply high-complexity components, supporting smooth RD-complexity tradeoffs and fine-grained adjustment (range of decoding times from 176 ms to 7283 ms, with BD-rate improvements over fixed baselines) (Gao et al., 2023).
Dynamic point cloud compression: Structured “slimmable” frameworks partition a network into multiple routes, each mapping to a distinct complexity-cost profile. A rate control module adaptively selects the encoding route per frame to hit target bitrate and complexity budgets, supporting frame-level variable route selection and per-route complexity measurement. BD-rate savings of 5.81%, BD-PSNR increases of 0.42 dB, and code time reductions up to 44.6% are reported, with average bitrate errors under 0.4% (Zhang et al., 28 Aug 2025).
Transform learning: While many learned transforms and entropy models are solely RD-optimized, some methods (e.g., RDLT) are designed to exploit the performance/computation frontier: data-driven, linear transforms are trained to minimize true RD loss using differentiable entropy models, providing improvements of up to 12% BD-rate over DCT in standard codecs, with the design intent of preserving linear, codec-friendly structure and limiting decoder complexity (Gnutti et al., 2024).

4. Theoretical Foundations and Algorithmic Advances

Seminal work in information theory and compression has been extended to consider complexity:

Layered/lower-complexity code constructions: Strong successive refinability theory shows that sources can be coded in layers without losing not only first-order but also second-order (source dispersion) optimality, in both the binary-Hamming and Gaussian-quadratic setting. Layered codes allow staged encoding with sub-exponential complexity $D$ 7, achieving the RD function with lower computational cost and explicit tradeoff expressions:

$D$ 8

Here, increasing the number of layers $D$ 9 lowers complexity but increases the rate penalty, highlighting the structural foundation of RDCO in finite-blocklength settings (No et al., 2015).

Semantic communication: Extensions of RD theory for semantic tasks introduce information-theoretic complexity constraints such as $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 0 (minimum description length, information bottleneck). Closed-form expressions for the minimum achievable rate under joint semantic distance and model complexity constraints are derived for both Gaussian and binary sources, formalizing a three-way tradeoff and providing an empirical basis to relate $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 1 to practical computational costs. This guides DNN encoder complexity selection in edge scenarios (Chai et al., 16 Feb 2026).
Rateless learned JSCC: In broadcast, variable-complexity joint source-channel coding with rateless schemes (NTRSCC) allows each receiver to stop reception and perform a user-specific number of BP decoding iterations. The total loss combines distortion, code rate, and BP decoder complexity, supporting practical, heterogeneous edge deployments (Qin et al., 23 Mar 2026).

5. Metric Design, Evaluation, and Application-Space Analysis

Traditional RD evaluation is insufficient in the RDCO context. New frameworks introduce 3D analytic and metric constructs:

Linear cost surface and application space: Systems are evaluated with $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 2, where $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 3 denote weights set by application-specific cost structure (e.g., streaming where bandwidth, quality, and hardware costs are mapped into monetary units). The codec selection problem becomes one of minimizing $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 4 at a specific point in the “application space.” Only codecs lying on the lower convex hull in RDC space can be optimal for some $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 5. In a study of 17 neural video codecs, only four were ever optimal for any application, with CR16 dominating for the typical streaming scenario $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 6 (Queiroz et al., 7 Sep 2025).
Generalized BD metrics and surface comparisons: To address the inadequacy of 2D BD-rate/PSNR for multi-objective tradeoffs, RDC-aware metrics project data onto planes or directions in RDC space parameterized by $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 7. However, due to cloud-like sample distributions and non-smooth surfaces for practical codecs, application-space Lagrangian cost is preferred for codec ranking.
Complexity quantification: Choices of complexity metric are explicit and platform-agnostic: single-thread CPU seconds (Zhong et al., 2021), kMAC/pixel (Queiroz et al., 7 Sep 2025), proportion of autoregressive decoding positions (Gao et al., 2023), or mutual information (Chai et al., 16 Feb 2026). This suggests metric definition is application-dependent and central to RDCO analysis.

6. Algorithmic and Structural RDCO Innovations

Given the computational intensity of high-dimensional or structural distortion measures, algorithmic innovation is central in RDCO:

Alternating mirror descent (AMD) for Gromov-type distortions: Expanding RD to structural distortions using the RDD function (rate distortion-in-distortion), the optimization involves quadratic or quartic costs over joint probability couplings. Efficient decomposition, linearization, and relaxation reduce the evaluation to $R(D) = \inf_{P_{Y|X}: \mathbb{E}[d(X,Y)] \le D} I(X;Y).$ 8 per iteration, making structural rate-distortion tradeoffs computationally feasible (Chen et al., 13 Jul 2025).
Communication optimal transport (CommOT): RD problems are framed as entropy-regularized optimal transport with one-sided marginal. Alternating Sinkhorn scaling and root finding achieve significantly faster computation of the rate-distortion function for high cardinality problems compared to Blahut–Arimoto methods, although without explicit optimization of a complexity term (Wu et al., 2022).

These developments directly address the complexity costs of more expressive or structurally robust coding schemes and open paths for future RDCO research.

7. Impact, Limitations, and Future Directions

RDCO has enabled new codec designs and system architectures that explicitly manage computational resources in conjunction with quality and bitrate targets. Practical outcomes include:

Dense and adjustable complexity ranges in video ladders (3%–100% of maximal preset, with corresponding RD savings) (Zhong et al., 2021).
Codec selection that adapts to real-world cost models and composite application needs (Queiroz et al., 7 Sep 2025).
Deployment of single-model codecs supporting a spectrum of compute-rate-distortion operating points for edge and streaming contexts (Gao et al., 2023, Zhang et al., 28 Aug 2025).
Principled extension of rate-distortion/semantic tradeoff theory to settings where encoder/decoder complexity is resource-constrained (Chai et al., 16 Feb 2026).

Challenges persist in establishing universally valid complexity measures, scaling optimization for high-dimensional or structural distortion objectives, and capturing system heterogeneity at deployment scale. The direction of research suggests further integration of theory from information bottleneck, optimal transport, and structural learning, and a growing emphasis on joint rate-distortion-complexity benchmarking in deployment-driven codec engineering.