Convex Optimization with Nested Evolving Feasible Sets

Summary

The paper introduces and studies Convex Optimization with Nested Evolving Feasible Sets (CONES). In this framework, the objective function f is fixed and known beforehand, but the feasible region evolves over time as a nested sequence of convex sets S 1 S 2 S T. At each time step t, an online algorithm must choose a feasible action x t in S t, aiming to simultaneously minimize the cumulative loss function cost C A(T) = sum t=1 T f(x t) and the total movement cost M A(T) = sum t=1 T x t - x t-1, where x 0 is a fixed initial action. The performance is compared against a static optimal benchmark x OPT = x in S T f(x), which is feasible for all sets due to nesting. The regret is defined as Regret A(T) = f, S t C A(T) - T f(x OPT). The paper characterizes the fundamental trade-off between regret and movement cost.

Key Contributions and Results:

1. Strongly Convex Case: When f is strongly convex, the paper first analyzes the greedy algorithm G (which always plays x t = x* t = x in S t f(x)). It proves that G has the minimum possible regret (non-positive) but incurs a movement cost of (sqrt T) (Theorems 2 and 3). To improve upon this, the paper proposes the F RUGAL algorithm (Algorithm 1). F RUGAL plays a lazy action (the projection of the previous action onto the current set) as long as its cumulative regret remains non-positive; otherwise, it performs a forced jump to the local minimizer. The paper shows that F RUGAL achieves non-positive regret (Lemma 1) and a movement cost of O(T) (Theorem 6), specifically M F RUGAL(T) = O (d d/2 2G + (L+ mu)D over mu T), for G-Lipschitz, L-smooth, and mu-strongly convex f. This is complemented by a lower bound (Theorem 8) showing that any online algorithm maintaining sub-linear regret O(t lambda) for lambda in [0,1) must incur a movement cost of (T), proving the optimality of F RUGAL.

2. α-sharp Case: The paper extends the analysis to alpha-sharp convex functions (Definition 9), which include linear functions. It shows that the F RUGAL algorithm again achieves non-positive regret and a movement cost of O(T) (Theorem 10), with a different proof technique that directly bounds the number of jumps. A matching lower bound (Theorem 11) is also established, proving that any algorithm with non-positive regret must have a movement cost of (T), thus proving F RUGAL's optimality in this case as well.

3. General Convex Case: When f is only convex, the paper shows a negative result: the greedy algorithm G has a movement cost of (T) (Theorem 12). To handle this general case, it proposes the Level Set Projection (LSP) algorithm (Algorithm 2), which tracks nested epsilon-sublevel sets. The LSP algorithm achieves a (regret, movement cost) tuple of (T 1-beta, T beta) for any beta in [0,1] (Theorem 13). Furthermore, for a structured input called directional (Definition 14), the paper proposes the G AP-F RUGAL algorithm (Algorithm 3), which achieves O(1) regret and O(squared T) movement cost (Theorem 15).

4. Linear Combination of Costs: The paper also considers the problem of minimizing a linear combination of loss and movement costs, sum t f t(x t) + sum t x t - x t-1 p, under dynamic constraints x t in S t. For p=1 and d=1, it proposes a 5-competitive algorithm (Theorem 16). For general d with alpha-sharp functions, it shows that the greedy algorithm G is 12 over alpha-competitive (Theorem 17).

The paper concludes that F RUGAL achieves the optimal trade-off for strongly convex and α-sharp functions, while the general convex case remains an open problem. It also discusses applications in margin-constrained classification and AI alignment, and compares its framework with sensitivity analysis, trajectory tracking, and model predictive control.

Based on the paper, here are specific improvements that can be made to AI systems, along with what the improved system can do.

1. For AI Alignment and Safe RLHF (Section 10.2)

• Improvement: Replace the current heuristic trust-region updates (which use fixed KL penalties) with the F RUGAL algorithm to manage policy updates. The algorithm decides when to make a large update (a jump) and when to make only minimal, lazy projections to stay within safety constraints.

• What the improved system can do: It can maintain a non-positive regret (i.e., it never falls behind the optimal policy's performance on the KL-regularized reward) while guaranteeing a total policy drift (movement cost) of only O(log T) over T training rounds. This directly prevents catastrophic forgetting and the alignment tax by ensuring the model doesn't oscillate wildly between updates, even when safety constraints (e.g., cost thresholds, KL budgets) are tightened adversarially. It provides a formal, worst-case guarantee on stability that current methods lack.

2. For Margin-Constrained Classification (Section 10.1)

• Improvement: Use the Level Set Projection (LSP) algorithm to handle progressively tightening margin constraints. Instead of re-optimizing the classifier from scratch each time the margin requirement increases, the system projects its current weights onto the new feasible set, only triggering a full re-optimization when the current solution becomes infeasible.

• What the improved system can do: It can achieve a tunable trade-off between classification error (regret) and model stability (movement cost). Specifically, for any β ∈ [0,1], it can achieve a regret of O(T(1-β)) and a movement cost of O(T β). This allows the system to be configured for either rapid adaptation (low regret, high movement) or high stability (high regret, low movement) depending on the application's needs, without requiring a full retraining at every constraint update.

3. For General Constrained Online Optimization (CONES)

• Improvement: Implement the G AP-F RUGAL algorithm for scenarios where the feasible set changes in a directional manner (a common, structured case). This algorithm adds a small tolerance to the regret condition, allowing for more lazy updates.

• What the improved system can do: For this structured input, it can achieve a constant regret (O(1)) and a movement cost of only O(log2 T). This is a significant improvement over the general convex case (which has a polynomial trade-off) and is nearly as good as the strongly-convex case, but it only requires the objective to be convex, not strongly convex. This is ideal for systems where constraints evolve predictably (e.g., gradually increasing safety thresholds).

4. For Online Optimization with Switching Costs (Problem 4)

• Improvement: For one-dimensional problems (d=1), replace the standard greedy approach with the AB algorithm described in Section 8.1. This algorithm explicitly balances the service cost (loss) against the movement cost by only moving to points where the movement cost is less than or equal to the service cost.

• What the improved system can do: It can achieve a 5-competitive ratio for the total cost (loss + movement) even when the feasible set changes over time. This provides a strong, constant-factor performance guarantee that is robust to adversarial constraint sequences, which is not achievable with naive greedy methods.

5. For High-Dimensional Problems with Sharp Loss Functions

• Improvement: Use the Greedy algorithm (G) when the loss function is α-sharp (e.g., linear functions). The paper proves that this simple algorithm has a bounded competitive ratio for the combined loss and movement cost.

• What the improved system can do: It can achieve a competitive ratio of 12/α for any dimension d. This provides a simple, computationally cheap, and provably robust solution for problems where the loss function has a sharp minimum (like linear or piecewise-linear objectives), guaranteeing that the total cost is within a constant factor of the optimal offline solution, even with dynamic constraints.

Convex Optimization with Nested Evolving Feasible Sets (CONES) is considered where the objective function f remains fixed but the feasible region evolves over time as a nested sequence S 1 S 2 S T. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing problem. When the loss function is convex, we propose a lazy-algorithm and show that it achieves O(T 1-beta), O(T beta) simultaneous regret and movement cost for any beta in (0,1], over a time horizon of T. When the loss function is strongly convex or alpha-sharp, we propose an algorithm Frugal that simultaneously achieves zero regret and a movement cost of O(T). To complement this, we show that any online algorithm with o(T) regret has a movement cost of for both cases, proving optimality of Frugal.