先看這個問題為什麼值得解Why this problem is worth solving
情境先行Start with the situation
地圖可以非常精確，但如果畫的是另一座城市，精確度幫不了你回家。定理的 assumptions 就是城市邊界。 A map can be precise yet useless if it depicts another city. Theorem assumptions define the city limits.
只看 rate 或 bound 大小，會忽略它是否適用於關心的資料、模型與步數。 Reading only rates or bound sizes ignores whether they apply to the data, model, and horizon of interest.
先備知識Prerequisite
- 看得懂 expectation、norm 與 big-O Can read expectations, norms, and big-O notation
讀完你能做到After this path
- 從定理抽出假設與保證 Extract assumptions and guarantees from a theorem
- 辨識漸近結果與有限範圍的差異 Distinguish asymptotic and finite regimes
- 設計會讓保證失效的邊界案例 Design an edge case that breaks the guarantee
閱讀框架Concept frame
- 核心問題Question
- 更緊的界線（bound），是否一定更接近實際深度學習？ Does a tighter bound necessarily say more about practical deep learning?
- 對照軸Comparison axis
- 實際 regime 與 theorem assumptions 的距離 Distance from theorem assumptions
- 判讀方式Evidence to inspect
- 並排閱讀三種相對情境，再回到論文檢查消融實驗、反事實測試與適用範圍。Read three relative scenarios side by side, then inspect the paper's ablations, counterfactual tests, and scope statements.
回答後查看解析Answer, then view the explanation
你的預測Make a prediction
比較兩個 convergence bound 時，第一步是什麼？ What is the first step when comparing two convergence bounds?
看現象，不急著講故事Observe before narrating
三個診斷訊號Three diagnostic signals
這些是待檢查的診斷假說。下一步用條件比較、消融實驗或反事實測試排除其他解釋。These are diagnostic hypotheses to test. Use condition comparisons, ablations, or counterfactual tests to rule out alternatives.
一次看完三種情境Read three scenarios side by side
三種情境對照Three-scenario comparison
哪個 assumption 最值得先放鬆，為什麼？ Which assumption should be relaxed first, and why?
對照軸：Comparison axis: 實際 regime 與 theorem assumptions 的距離 Distance from theorem assumptions
可直接測 rate、常數與 tightness。 Rates, constants, and tightness can be tested directly.
應逐項做 assumption stress test。 Stress-test assumptions one at a time.
定理主要提供機制直覺，不應當成數值預測。 Treat the theorem as mechanism intuition, not a numerical forecast.
從直覺到機制From intuition to mechanism
把機制講清楚Explain the mechanism
定理是一份有條件的契約：條件成立，就保證某件事；不是無條件預言。 A theorem is a conditional contract: meet the assumptions and receive a guarantee—not an unconditional prophecy.
輸入是問題設定與假設；輸出是形式化保證；目標是刻畫可能性、極限或機制。 Input: a problem setting and assumptions. Output: a guarantee. Objective: characterize possibility, limits, or mechanism.
- 01 列出資料、模型、loss 與 algorithm assumptions。 List data, model, loss, and algorithm assumptions.
- 02 確認 guarantee 的 metric、機率與時間尺度。 Identify the guarantee's metric, probability, and horizon.
- 03 找 tight example、lower bound 或 assumption violation。 Find tight examples, lower bounds, or assumption violations.
把 asymptotic rate 的改善直接翻譯成有限算力下的速度。 Translating a better asymptotic rate directly into finite-compute speed.
四格閱讀法：assumptions、guarantee、regime、failure case。 Use four boxes: assumptions, guarantee, regime, and failure case.
不用回看頁面的自我檢查Check your recall without looking back
三個回想題Three recall questions
先口頭回答，再展開解析。Answer aloud before opening each explanation.
01 vacuous bound 是什麼？ What is a vacuous bound?
形式上正確，但在關心的參數範圍沒有提供有用限制。 A formally correct bound that gives no useful restriction in the relevant regime.
02 為何常數不能總被 big-O 忽略？ Why can constants not always be ignored in big-O?
有限資料與有限步數下，常數可能主導實際大小。 At finite data and horizons, constants may dominate.
03 assumption stress test 怎麼做？ How do you stress-test an assumption?
固定其他條件，逐步違反該 assumption，觀察 guarantee 對應現象是否消失。 Hold others fixed, violate one assumption progressively, and observe whether the guaranteed behavior disappears.
把工具帶回你的研究Take the tool back to your research
最後一個任務One final task
從一篇理論論文挑主定理，做一張四格卡，再寫一個會讓其中一項 assumption 失效的資料例子。 Take a main theorem, fill a four-box card, then construct data that violates one assumption.