End-to-End Learning Guarantee

• End-to-End Learning Guarantee is a formal framework ensuring that integrated ML systems provide certifiable guarantees on safety, stability, and optimality from raw, high-dimensional inputs.
• Methodologies incorporate calibration properties, Lyapunov-based stability, and robust MILP-based verification to jointly address prediction, control, and optimization challenges.
• Empirical implementations in adaptive control, vision-based driving, and safe RL demonstrate effectiveness, even as challenges in scalability and compositional verification persist.

An end-to-end learning guarantee establishes formal criteria under which a machine learning system, configured to optimize decisions or control actions directly from high-dimensional raw input (e.g., sensor or perception data), provides certifiable guarantees with respect to global task objectives such as safety, stability, optimality, or robustness. Unlike traditional modular machine learning pipelines—where prediction and decision-making can be separately analyzed—end-to-end learning entangles the entire data→model→decision stack, raising fundamental challenges for formal verification and theoretical risk analysis.

1. Formal Definitions and Theoretical Frameworks

End-to-end learning guarantees manifest in various domains through distinct but related notions of certification, such as Lyapunov-based stability for control, Fisher consistency/calibration for predict-then-optimize pipelines, game-theoretic stability for auction mechanisms, or robust risk upper bounds under adversarial perturbations.

A canonical example in end-to-end prediction–optimization is as follows (Ho-Nguyen et al., 2020):

• Given side-information $w$ and unknown cost $c$, one estimates parameters $d = g(w)$ using a predictor $g$ and solves $\min_{x \in X} f(x) + d^T x$ to select $x^*(d)$. The performance is assessed by the true optimality gap $$L(d, c) = f(x^*(d)) + c^T x^*(d) - \min_{x \in X} f(x) + c^T x$$.
• The guarantee framework asks: under what conditions on the surrogate loss $\ell$, does minimizing expected surrogate risk $R_\ell(g) = \mathbb{E}[\ell(g(w), c)]$ induce low expected true risk $R(g) = \mathbb{E}[L(g(w), c)]$? The answer is pointwise Fisher consistency or calibration: near-minimizers of $\mathbb{E}[\ell(\cdot, c) \mid w]$ are near-minimizers of $\mathbb{E}[L(\cdot, c) \mid w]$.
• Stronger, uniform calibration properties yield explicit nonasymptotic bounds: for instance, with squared loss on compact $X$, $R(g) - R^* \leq B_X \sqrt{R_{LS}(g) - R_{LS}^*}$ (Ho-Nguyen et al., 2020).

For safety-critical control, guarantees typically take the form of Lyapunov-based stability proofs for the closed-loop system under the learned (possibly neural) control law, extended to the online adaptive setting (Ryu et al., 6 Mar 2024, Wang et al., 2023).

In robust end-to-end optimization pipelines, certification may require integrating both ML and downstream optimization model uncertainty into a unified min–max robust risk, whose relaxation can exactly bound the worst-case realized task loss under bounded perturbations (Xu et al., 2023).

2. Domain-Specific Methodologies for End-to-End Guarantees

End-to-End Control with Formal Tracking/Stability Guarantees

CNN-based adaptive controllers are constructed to map historical sequences of errors, states, and controls into current control actions. The controller parameters are updated online via projected, damped gradient descent, with the adaptation law carefully structured to allow Lyapunov analysis (Ryu et al., 6 Mar 2024). The main result establishes that, under smoothness and realization assumptions, and provided the control gain is chosen appropriately, the tracking error $e(t)$ asymptotically converges: $e(t) \rightarrow 0$ as $t \rightarrow \infty$, with all controller weights remaining bounded.

The control guarantee is anchored by:

• Explicit parameterization of the closed-loop error dynamics,
• Definition of a composite Lyapunov candidate $V(e, \tilde\theta)$,
• Formal derivative analysis under the adaptive update,
• Admissible control-gain selection $k_s \geq \beta_1 \beta_2^2 + \bar{\Delta}$ ensuring negativity of the Lyapunov derivative and convergence by Barbalat's lemma.

Empirical evidence shows that such CNN-based controllers robustly outperform DNNs without explicit temporal feature extraction when faced with plant variations or modeling uncertainties (Ryu et al., 6 Mar 2024).

Stability Certification in End-to-End Perception-to-Control

End-to-end vision-based driving policies use observation-parameterized Lyapunov functions (stability attention CLFs, or att-CLFs) embedded in differentiable optimization layers (Wang et al., 2023). The task-conditional Lyapunov function adapts the stabilization priorities based on visual context, with robust enforcement via QP constraints at every timestep. Theorems are provided for:

• Classical CLF exponential stability for affine systems.
• Partial-state exponential stability for systems controlled via att-CLFs, with adaptive, observation-dependent weighting of stabilization directions.
• Extensions include uncertainty propagation through MC sampling, yielding probabilistically robust controls under perceptual noise, thus closing the loop for stability guarantees in the presence of learned perception.

Robust Certification in Predict–Then–Optimize Systems

Robust end-to-end learning frameworks treat both feature-space (input) and downstream optimization (CO) uncertainties jointly. The robustified objective is: $$\min_{\theta} \mathbb{E}_{(x, y, \phi)} \left[ \max_{\delta_x \in \Delta_x,\, \delta_\phi \in \Delta_\phi} L(z^*(f(x+\delta_x; \theta), \phi + \delta_\phi); y, \phi + \delta_\phi) \right]$$ where $z^*$ is the argmin solution to the CO problem (Xu et al., 2023). Certification is delivered by solving an associated MILP, whereby for every sample, no admissible (bounded) perturbation can increase the task loss beyond a computed certificate value. This result holds rigorously whenever the ML model and CO admit mixed-integer linear or convex quadratic reformulations.

Failure to properly integrate CO-stage uncertainty during training can open new generalization gaps not present in conventional ML robustness theory.

Game-Theoretic Incentives and End-to-End Mechanism Learning

In end-to-end neural auction mechanisms, the learning guarantee extends to economic desiderata such as incentive compatibility and individual rationality. By parameterizing neural rankers with architectures admitting monotonicity constraints and critical-price computation (e.g., MIN-MAX networks), and integrating relaxations of naturally discrete operations (e.g., sorting via NeuralSort), such frameworks achieve ex-post IC and IR, measured by near-zero empirical regret metrics (Liu et al., 2021).

3. Safety and Robustness: Constraints, Adversaries, and Lifetime Guarantees

In reinforcement learning and cyber-physical applications, end-to-end lifetime safety is an open challenge. Provably Lifetime Safe RL (PLS) integrates offline return-conditioned policy learning (e.g., constrained Decision Transformers) with online tuning of a low-dimensional “target return” via safe Gaussian process optimization (Wachi et al., 28 May 2025). The main high-probability guarantee (Theorem 4.1) ensures that all policies deployed during learning and operation respect hard safety constraints with probability $1-\Delta$: $$\forall i,\; J_g(\pi_{z_i}) \leq b \text{ with probability } \ge 1 - \Delta$$ PLS achieves this by maintaining a certified safe set and applying stagewise safe Bayesian optimization over returns, as justified by asymptotic GP regression theory (Theorem 3.1).

Formal verification of end-to-end learning policies remains hampered by strong modeling assumptions: symbolic-state accessibility, complete agent behavior models, and exact kinematics (Fulton et al., 2020). Ongoing work seeks to relax these via intermediary template-matching perception models, online falsification, and verification-preserving program synthesis, but fully composable end-to-end safety proofs in the presence of rich learned components are not yet available.

4. Technical Assumptions and Calibration Properties

End-to-end guarantees often hinge on calibration or consistency properties—either of losses (Fisher consistency, pointwise or uniform calibration) or models (realizability, boundedness, smoothness, Hurwitz stability). For example, in predict–then–optimize pipelines:

• The squared loss achieves exact uniform calibration, leading to explicit nonasymptotic risk bounds (Corollary 1 in (Ho-Nguyen et al., 2020)).
• Surrogates lacking Fisher consistency (e.g., multiclass hinge under some $P$) can yield suboptimal true decision risk, regardless of the surrogate risk minimization performance.

In adaptive control, convergence and boundedness critically require the existence of ideal network parameters within prescribed bounds (Assumption A3 in (Ryu et al., 6 Mar 2024)), as well as Hurwitz-ness of design matrices controlling the error dynamics.

Robustness gains in adversarial E2E learning depend on the representability of the predictor and CO in MILP or QP forms, and on the underlying generalization-capacity of the ML model class.

5. Empirical Demonstrations and Practical Implications

Empirical evaluation of end-to-end certified learning commonly proceeds along the following lines:

• In adaptive control (Ryu et al., 6 Mar 2024), CNN-based controllers achieve lower RMSE than DNNs, and display faster adaptation in the presence of plant changes.
• In robust E2E learning (Xu et al., 2023), end-to-end adversarially trained models maintain certified performance under bounded feature and model perturbations—formalized and numerically attained via mixed-integer programming.
• In safe RL (Wachi et al., 28 May 2025), PLS outperforms alternative baselines, achieving strictly zero safety violations across safety-constrained tasks (Safety-Gym/Bullet), while matching or exceeding state-of-the-art rewards.
• For formally certified federated learning (Lee et al., 19 Apr 2024), aggregate model updates can be provably linked, via zkSNARK proofs and blockchain verification, to (only) legitimate, attested local computation and authenticated device data, ensuring computational integrity across the learning workflow.

Guarantee Type	Methodology	Domain	Key Assumptions
Lyapunov stability	Online adaptive control (CNN)	Nonlinear plant control	Smoothness, Hurwitz matrix, ideal parameter in bounds
Task–loss calibration	Surrogate risk minimization	Predict–optimize pipelines	Fisher/uniform calibration of surrogate loss
Robustness certification	Adversarial E2E training	ML+optimization tasks	Piecewise linear predictor, QP constraints
Lifetime safety	Safe RL + GPs (PLS)	RL under constraints	Model expressivity, safe offline data, Lipschitzness
Economic stability (IC/IR)	Neural mechanism design	Auction markets	Monotonic rankers, critical-price computation
Computational integrity	zkSNARK + blockchain	Decentralized FL	Secure crypto, binding commitments, sound circuits

6. Limitations and Ongoing Challenges

Despite progress, several limitations are evident:

• Full formal verification of end-to-end pipelines combining learned perception, estimation, and control under realistic (non-symbolic, data-driven) environments remains open due to compounded uncertainty and lack of symbolic world models (Fulton et al., 2020).
• End-to-end guarantees can degrade if key technical conditions (e.g., Fisher consistency, proper surrogate risk minimization, or model expressivity) are violated in practice.
• Uniform end-to-end certification via exact MILP or convex optimization is computationally tractable only for low- to moderate-dimensional problems. Scalability to high-dimensional vision–control tasks requires approximate, probabilistically justified relaxations.
• Empirical gap between theoretical guarantees and real-world deployment exists, often due to non-idealities not captured in formal proofs (e.g., sensor shifts, adversarial attacks, changing agent behaviors).

A plausible implication is that further advances in compositional verification, robust learning under partial observability, and unifying frameworks spanning task-aware surrogate design, adversarial certification, and formal control are necessary for universal, domain-independent end-to-end guarantees.